The Pfaffian product conjecture for the quasi-eigenfunction at $\alpha=q^{1/2}t^{1/2}$

Let $F(\alpha)$ be the homogeneous quasi-eigenfunction in $n$ variables, and let $n$ be a positive even integer.

Pfaffian product conjecture. At $\alpha=q^{1/2}t^{1/2}$,

$$F(q^{1/2}t^{1/2})=\operatorname{Pfaffian}\left(\frac{1-\zeta_j^2/\zeta_i^2}{(1-q^{-1/2}t^{1/2}\zeta_j/\zeta_i)(1-q^{1/2}t^{-1/2}\zeta_j/\zeta_i)}\right)_{1\leq i,j\leq n} \prod_{1\leq i<j\leq n}\frac{(q^{1/2}t^{-1/2}\zeta_j/\zeta_i;q)_\infty}{(q^{1/2}t^{1/2}\zeta_j/\zeta_i;q)_\infty}.$$

For odd $n$, the formula is obtained by taking the limit $\zeta_n\to0$.

The formula is presented as an observed factorization and is supported by the low-rank calculations in the paper. General validity is conjectural.