Generalized geography on solid grid graphs
Just to complete my comment:

GG remains PSPACE-complete on planar bipartite directed graphs with maximum degree 3 (see
D. Lichtenstein, M. Sipser: GO Is Polynomial-Space Hard. J. ACM 27(2): 393-401 (1980) )

But (vertex) GG on undirected graphs is in P (see this Q&amp;A and this paper) so it is in P also for grid graphs.

For directed solid grid graphs I didn't find any reference, however, I think there is an easy way to simulate a planar bipartite directed graph; the following idea should work:


Both the "diamond" structure (A) and the crossover gadget (C), can be converted into an equivalent (directed) solid grid graph gadget (B) and (D).