Note: I have posted a hugely expanded version of this question on cstheory.
Since a Sokoban instance with only $k$ boxes has at most $n^{O(k)}$ possible states, the problem lies in $\operatorname{NSPACE}$$[O(k \cdot \log (n))]$. Since the region
X X
X XX XX
X X X
XX X
XXX
has neither boxes nor storage locations and is such that a box can be pushed from the top to exit out the bottom but can't be pushed from the bottom to the top, the problem "Is there a matching from one given set of $k$ vertices to another given set of $k$ vertices in the transitive closure of the given directed grid graph?" can easily be reduced to the problem I am asking about.
Is anything else known about the complexity of Sokoban with only $k$ boxes?
In particular, is it the case that solvable instances always have solutions of length $n^{o(k)}$?
I tried to construct a bounded-counter gadget, but aside from realizing that such a gadget would require a box being moved along it when the counter's value is sufficiently large (since otherwise the player could just "pretend" to have incremented or decremented the counter), I wasn't able to get anywhere with that.
