Behrend–Fischer–Koutschan symmetry conjecture for off-diagonally symmetric ASMs

Let $\text{OSASM}(2n)$ denote the set of off-diagonally symmetric alternating sign matrices of order $2n$. For $X\in\text{OSASM}(2n)$, let $\mathsf{R}(X)$ be the number of nonzero strictly upper-triangular entries and let $\mathsf{T}(X)$ be the column containing the $1$ in the first row. Define

$$S(2n;\rho,\tau)=\{X\in\text{OSASM}(2n):\mathsf{R}(X)=\rho,\mathsf{T}(X)=\tau\}.$$

Behrend–Fischer–Koutschan symmetry conjecture. Given $n$, for any $\rho$ and $\tau$, we have

$$|S(2n;\rho,\tau)|=|S(2n;\rho,2n+2-\tau)|.$$

This conjecture asserts a reflection symmetry in the refined enumeration of off-diagonally symmetric alternating sign matrices according to the number of nonzero strictly upper-triangular entries and the position of the first-row $1$. Its resolution is not indicated in the supplied text.