The torsion-freeness and rank conjecture for square polyominoes

Let $C_4$ denote a square cycle, and let $G^4_S$ be a square polyomino formed from $S$ copies of $C_4$. Write $V(G^4_S)$ and $E(G^4_S)$ for its vertex and edge sets, and let $\operatorname{MH}_{k,k}(G^4_S)$ denote its magnitude homology on the main diagonal. Square-polyomino conjecture. The main diagonal of the magnitude homology of a square polyomino is torsion-free and satisfies

$$\operatorname{rank}(\operatorname{MH}_{k,k}(G^4_S)) \cong \begin{cases} |V(G^4_S)| & k=0,\\ 2|E(G^4_S)|+4(i-1)S & k\geq 1.\end{cases}$$

This is known for polyominos constructed exclusively using Type I moves, while the Type II case remains open because the required Mayer–Vietoris argument is unavailable.