The determinant evaluations for symmetric alternating-sign-matrix enumerators

Let $r$ be a positive integer. Let $R_r^{\mathrm{o}}$, $C_r^{\mathrm{e}}$, and $C_r^{\mathrm{o}}$ be the matrices defined in Corollary~, and let $A_k^{\mathrm{VS}}(t)$ and $A_k^{\mathrm{HTS}}(t)$ denote the corresponding refined alternating-sign-matrix enumerators.

Determinant-evaluation conjecture. The following identities should hold:

$$\det R_r^{\mathrm{o}}(t)=A_{2r+1}^{\mathrm{VS}}(t),$$ $$\det C_r^{\mathrm{e}}(t)=A_{2r}^{\mathrm{HTS}}(t),$$ $$\det C_r^{\mathrm{o}}(t)=A_{2r-1}^{\mathrm{HTS}}(t).$$

These identities are the remaining determinant-evaluation problem after the paper derives determinant formulas for the relevant generating functions. The matrices are only defined by reference to the source's corollary, so the row should be checked against that definition.