Complete maximal identity conjecture for Cartesian products of paths

Let $P_i$ denote the path graph with $i$ vertices, and let $\square$ denote the Cartesian product of graphs. For $i\leq j$ with $i,j\in\mathbb{Z}_{>1}$, consider $P_i\square P_j$. A graph has the complete maximal identity property if its maximal stable configuration is equivalent to the identity for every choice of sink.

Path-product conjecture. The only graphs $P_i\square P_j$ with the complete maximal identity property are $P_2\square P_j$ when $j\equiv1\pmod{3}$, together with $P_2\square P_2$, which is $K_4$.

The conjecture was supported by a SageMath verification for $1<i\leq j\leq100$. The proposition establishing the listed $P_2\square P_j$ cases is proved in the paper, but the exclusion of all other path products remains open.