Even-order off-diagonally symmetric alternating sign matrix generating-function symmetry conjecture

Let $X^{\mathrm O}_{2n}(r,t)$ be the even-order OSASM generating function, and let $\bar t$ denote the parameter appearing in the source's generating-function notation. For an OSASM $A$, let $R(A)$ and $T(A)$ denote the statistics used to define this generating function. Even-order OSASM symmetry conjecture. The generating function satisfies

$$X^{\mathrm O}_{2n}(r,t)=t^{2n+2}X^{\mathrm O}_{2n}(r,\bar t).$$

Equivalently, for every $\rho$ and $\tau$,

$$|\{A\in\mathrm{OSASM}(2n):R(A)=\rho,\ T(A)=\tau\}| = |\{A\in\mathrm{OSASM}(2n):R(A)=\rho,\ T(A)=2n+2-\tau\}|.$$

The conjecture also asserts the coefficient identity

$$[t^3]X^{\mathrm O}_{2n}(r,t)=[t^{2n-1}]X^{\mathrm O}_{2n}(r,t)=r\bigl((n-2)r^2+1\bigr)X^{\mathrm O}_{2n-2}(r,1).$$

These claims describe a proposed symmetry of the refined enumeration of even-order OSASMs; the paper presents them as conjectural and supplies no proof.