The Igusa-function formula conjecture for the middle symplectic local zeta function

For a positive integer $n$, let $q$ be the residue-field cardinality and let $s$ be a complex variable. Let $\mathcal{Z}_{n,n,\mathfrak{o}}^{\mathsf{C}}(s)$ denote the symplectic local zeta function at the middle parameter $\ell=n$, and let

$$\operatorname{I}_n(Y;X_1,\dots,X_n)=\sum_{I\subseteq[n]}\binom{n}{I}_Y\prod_{i\in I}\frac{X_i}{1-X_i}$$

be the Igusa function of degree $n$. The Igusa-function formula conjecture. One has

$$\mathcal{Z}_{n,n,\mathfrak{o}}^{\mathsf{C}}(s)=\frac{1}{1-q^{\binom{n+1}{2}-ns}}\operatorname{I}_n\left(q^{-1};\left(q^{\binom{n+1}{2}-\binom{i}{2}-ns}\right)_{i\in[n]}\right).$$

This is motivated by the corresponding established formula for $\ell=0$ and by computations for $n\leqslant 13$; the claimed identity for the middle value $\ell=n$ remains conjectural in the supplied text.