Equitable partitions of rectangular grid graphs

Let $P_m\oblong P_n$ be the rectangular grid graph, and let its automorphism group act naturally on its vertices. An equitable partition is a partition in which every vertex in a cell has the same number of neighbors in each cell.

Grid-graph equitable-partition conjecture. The equitable partitions of $P_m\oblong P_n$ are the orbit partitions of natural actions of the subgroups of the automorphism group of the graph. The lattice of equitable partitions is isomorphic to the lattice of subgroups exactly when there is a vertex with trivial stabilizer subgroup. This happens unless $2\le m=n\le3$.

The claim concerns the relationship between equitable partitions and symmetry-induced orbit partitions for rectangular grid graphs. The supplied text reports this as an apparent pattern based on computations, but gives no resolution or proof.