The conjectural terminating elliptic hypergeometric sum for the $A_n$ root system

Let $N$ be a positive integer, let $t$ and $t_1,\dots,t_{n+3}$ be parameters, and suppose that

$$\prod_{k=1}^n t_k=q^{-N}.$$

For an integer $\lambda$, write $\theta(a)_\lambda=\theta(a;p;q)_\lambda$. The terminating $A_n$ elliptic hypergeometric sum conjecture. The following identity should hold:

$$\sum_{\substack{\lambda_k=0,\ldots,N\\lambda_1+\cdots+\lambda_n=N}} \frac{\prod_{1\leq i<j\leq n}\theta(tt_it_j)_{\lambda_i+\lambda_j}\prod_{i=1}^n\prod_{j=n+1}^{n+3}\theta(tt_it_j)_{\lambda_i}\prod_{i,j=1}^n\theta(t_it_j^{-1})_{-\lambda_j}} {\prod_{i,j=1;\ i\neq j}^n\theta(t_it_j^{-1})_{\lambda_i-\lambda_j}\prod_{j=1}^n\theta(t^{n+1}t_j^{-1}\prod_{k=1}^{n+3}t_k)_{-\lambda_j}} = \begin{cases} \displaystyle\frac{\theta(1)_{-N}}{\theta(t^{n/2})_{-N}\prod_{n+1\leq i<j\leq n+3}\theta(t^{(n+2)/2}t_it_j)_{-N}},&n\text{ is even},\\[6pt] \displaystyle\frac{\theta(1)_{-N}}{\prod_{i=n+1}^{n+3}\theta(t^{(n+1)/2}t_i)_{-N}\theta(t^{(n+3)/2}\prod_{i=n+1}^{n+3}t_i)_{-N}},&n\text{ is odd}. \end{cases}$$

The conjecture is presented as an elliptic extension of a previously known theorem and concerns sums of residues of derived $A_n$ integrals. Its resolution status is not specified in the source.