Mills–Robbins–Rumsey's refined vertical-symmetry conjecture

Let $n\geq1$, let $1\leq r\leq2n-1$, let $\TSPP{2n+1}$ be the triangular shifted plane partitions defined above, and let $\gamma$ be the odd-diagonal flip involution. Write $U_2(b)$ for the statistic defined above, let $\TSPP{2n+1}^{\gamma}$ be the set of $\gamma$-invariant elements, and let $A_{2n+1}^{\mathrm{VS}}(t)$ denote the refined enumerator of vertically symmetric alternating sign matrices.

Mills–Robbins–Rumsey's refined vertical-symmetry conjecture. The number of $b\in\TSPP{2n+1}$ satisfying $\gamma(b)=b$ and $U_2(b)=r-1$ should equal the number of vertically symmetric alternating sign matrices satisfying the stated first-column condition $a_{i1}=1$; equivalently,

$$\sum_{b\in\TSPP{2n+1}^{\gamma}}t^{U_2(b)}=A_{2n+1}^{\mathrm{VS}}(t).$$

The conjecture gives a refined enumeration for the vertical-symmetry class. The paper proves a determinant formula for the left-hand generating function, but the claimed equality with the alternating-sign-matrix polynomial is presented as conjectural.