Square-free Smith forms for q-weighted lozenge-tiling matrices

For the explicitly defined $q$-weighted Kasteleyn and Kasteleyn-Percus matrices $M(a,b,c;q)$, $M_\rho(a,a,a;q)$, $A_\tau(a,b,b;q)$, $\widetilde{A}_\tau(a,b,b;q)$, and $\widetilde{A}_{\langle\rho,\tau\rangle}(a,a,a;q)$, take Smith normal forms over $\mathbb{Z}[q,q^{-1}]$. A Smith normal form is square free when its nontrivial invariant factors are square-free in the relevant Laurent polynomial ring. Square-free Smith-form conjecture. These five Smith normal forms are square free. Together with the roundness conjecture, this would determine the corresponding forms from their determinants and would address the cited Kasteleyn-cokernel problem for hexagonal regions. The paper notes that the claim is not fully general and gives a family where square-freeness fails; the stated cases remain open.