I still have strong doubts that this would lead to anything useful, given the low search depth the I.D. will be able to reach. One problem with drop variants is that the drop moves are very prone to causing horizon effect. In face of an unsolvable threat it is almost always possible to make a sacrificial drop of a low-valued piece that attacks a piece of much higher value, which cannot be ignored. But it can be dealt with by simply capturing the dropped piece. This necessary interlude will then have burned 2 ply, and will of course not have solved anything; the AI will still suffer the inescapable loss, but doesn't search deep enough anymore to see it. And the loss it is still within the horizon, rinse and repeat to sacrifice more material for pointless delay...
Without drops this occurs only occasionally. Because low-valued pieces typically are low-valued because they have low mobility, and thus are unlikely to have the possibility to attack a high-valued piece, even in an unsound manner. While pieces that habitually would have this possibility are high-valued themselves, so that sacrificing those comes out worse than accepting the loss you try to delay. But the drop rule gives low-valued pieces unlimited mobility when they are in hand.
So I wonder if it wouldn't be better to let the AI search drops based on strategic considerations, rather than tactical ones. Being on the board severly limits the number of empty squares that can be reached, compared to being in the hand. Therefore pieces in hand are usually considered worth somewhat more than on the board. OTOH, pieces on the board can have captures, or interdict enemy access to empty squares ('board control'). This board control, especially over squares in your own camp and near your own King is a quite valuable strategic aspect. One way to quantify it is give points for attacked squares, the lower the value of the lowest-valued attacker, the more points. Extra attacks on an already controlled square could be worth something, but much less. It would be much better to have two different squares defended by a Pawn, than aim two Pawns at the same square, and leave the other square undefended.
The value of board control could be used to tune the tendency to hoard pieces in the hand. They are intrinsically worth more there (but the bonus should be lower for a second piece of the same type). But having too many pieces in hand would mean you have to few on the board for effective board control, and this must be so costly that repairing it would overcome the intrinsic value loss on dropping. Attacking enemy pieces should be even more valuable than controlling empty squares (which as a collateral would make dropping on squares where you are attacked less valuable), which would bias the drops to attacking ones rather than defensive ones. (Having a piece that attacks an equal-valued piece is almost as good as having that piece in hand, as you can trade it when you need to drop, creating the dilemma for the opponent to either allow the drop or refrain from recapture.)
By not considering strategically unattractive drops (i.e. on insufficiently controlled squares) when little depth is remaining you don't only eliminate nodes (and thus gain affordable depth), but you prevent this horizon effect. And it would still consider strategically sensible drops there, speculatively awarding the good they might do beyond the horizon. So there will still be a chance that a drop searched at reduced depth will get a good score, and as a consequence be re-examined to investigate whether it can tactically deliver on its strategic promise.
I still have strong doubts that this would lead to anything useful, given the low search depth the I.D. will be able to reach. One problem with drop variants is that the drop moves are very prone to causing horizon effect. In face of an unsolvable threat it is almost always possible to make a sacrificial drop of a low-valued piece that attacks a piece of much higher value, which cannot be ignored. But it can be dealt with by simply capturing the dropped piece. This necessary interlude will then have burned 2 ply, and will of course not have solved anything; the AI will still suffer the inescapable loss, but doesn't search deep enough anymore to see it. And the loss it is still within the horizon, rinse and repeat to sacrifice more material for pointless delay...
Without drops this occurs only occasionally. Because low-valued pieces typically are low-valued because they have low mobility, and thus are unlikely to have the possibility to attack a high-valued piece, even in an unsound manner. While pieces that habitually would have this possibility are high-valued themselves, so that sacrificing those comes out worse than accepting the loss you try to delay. But the drop rule gives low-valued pieces unlimited mobility when they are in hand.
So I wonder if it wouldn't be better to let the AI search drops based on strategic considerations, rather than tactical ones. Being on the board severly limits the number of empty squares that can be reached, compared to being in the hand. Therefore pieces in hand are usually considered worth somewhat more than on the board. OTOH, pieces on the board can have captures, or interdict enemy access to empty squares ('board control'). This board control, especially over squares in your own camp and near your own King is a quite valuable strategic aspect. One way to quantify it is give points for attacked squares, the lower the value of the lowest-valued attacker, the more points. Extra attacks on an already controlled square could be worth something, but much less. It would be much better to have two different squares defended by a Pawn, than aim two Pawns at the same square, and leave the other square undefended.
The value of board control could be used to tune the tendency to hoard pieces in the hand. They are intrinsically worth more there (but the bonus should be lower for a second piece of the same type). But having too many pieces in hand would mean you have to few on the board for effective board control, and this must be so costly that repairing it would overcome the intrinsic value loss on dropping. Attacking enemy pieces should be even more valuable than controlling empty squares (which as a collateral would make dropping on squares where you are attacked less valuable), which would bias the drops to attacking ones rather than defensive ones. (Having a piece that attacks an equal-valued piece is almost as good as having that piece in hand, as you can trade it when you need to drop, creating the dilemma for the opponent to either allow the drop or refrain from recapture.)
By not considering strategically unattractive drops (i.e. on insufficiently controlled squares) when little depth is remaining you don't only eliminate nodes (and thus gain affordable depth), but you prevent this horizon effect. And it would still consider strategically sensible drops there, speculatively awarding the good they might do beyond the horizon. So there will still be a chance that a drop searched at reduced depth will get a good score, and as a consequence be re-examined to investigate whether it can tactically deliver on its strategic promise.