q=-1 relations between Kasteleyn cokernels and Smith forms

Let $G$ and $G'$ be the symmetry groups specified in the claim, and write $M_G(a,b,c)_{-1}$ and $A_G(a,b,c)_{-1}$ for the corresponding matrices specialized at $q=-1$. Let $\operatorname{Sm}(N)$ denote the Smith normal form of $N$. $q=-1$ cokernel conjecture. The following relations should hold:

$$\operatorname{Sm}(A_{G'}(a,b,c))=\operatorname{Sm}(M_G(a,b,c)_{-1})^2$$

when $G=\langle1\rangle$ or $\langle\rho\rangle$ and $G'=\langle G,\kappa\rangle$;

$$\operatorname{coker} A_G(a,b,c)_{-1}\cong\operatorname{coker} M_{G'}(a,b,c)^{\oplus2}$$

when $G=\langle\tau\rangle$ or $\langle\rho,\tau\rangle$ and $G'=\langle\kappa\tau\rangle$ or $\langle\rho,\kappa\tau\rangle$, respectively; and

$$\operatorname{coker} A_{G'}(a,b,c)\cong\operatorname{coker} A'_G(a,b,c)_{-1}$$

when $G=\langle\tau\rangle$ or $\langle\rho,\tau\rangle$ and $G'=\langle G,\kappa\rangle$. This conjecture proposes a cokernel and Smith-form extension of Stembridge's $q=-1$ phenomenon for symmetry classes of plane partitions. The source presents it as conjectural and gives no proof or resolution, so it remains open.