The conjectural elliptic beta integral for the $A_n$ root system

Let $n$ be a positive integer. Let $z_i$, $i=1,\dots,n$, $t_k$, $k=1,\dots,n+1$, and $f_j$, $j=1,\dots,n+2$, be independent complex variables, and set

$$A=\prod_{k=1}^{n+1}t_k,\qquad B=\prod_{j=1}^{n+2}f_j.$$

With $z_1z_2\cdots z_{n+1}=1$, define

$$\Delta^I(\mathbf{z};A_n)=\frac{1}{(2\pi i)^n}\frac{\prod_{k=1}^{n+1}\left(\prod_{i=1}^{n+1}\Gamma(t_iz_k^{-1})\prod_{j=1}^{n+2}\Gamma(f_jz_k)\right)}{\prod_{i,j=1;\ i\neq j}^{n+1}\Gamma(z_iz_j^{-1})\prod_{k=1}^{n+1}\Gamma(ABz_k)}.$$

Suppose that $|t_k|,|f_j|<1$ and $|pq|<|AB|$. The $A_n$ elliptic beta integral conjecture. The following integration formula should hold:

$$\int_{\mathbb{T}^n}\Delta^I(\mathbf{z};A_n)\frac{dz_1}{z_1}\cdots\frac{dz_n}{z_n}=\frac{(n+1)!}{(q;q)_\infty^n(p;p)_\infty^n}\frac{\Gamma(A)\prod_{j=1}^{n+2}\Gamma(f_j^{-1}B)\prod_{k=1}^{n+1}\prod_{j=1}^{n+2}\Gamma(t_kf_j)}{\prod_{k=1}^{n+1}\Gamma(t_kB)\prod_{j=1}^{n+2}\Gamma(f_j^{-1}AB)}.$$

This conjecture proposes an elliptic beta integral evaluation associated with the $A_n$ root system and is intended as a tool for deriving further nontrivial $A_n$ integrals. Its resolution status is not specified in the source.