Warnaar's $tilde A_n$ basic hypergeometric summation conjecture

Let $n$ and $m$ be positive integers, let $M_1$ and $M_2$ be nonnegative integers, and let $S_1$ and $S_2$ be integers satisfying $-M_1\le S_1\le M_2$ and $-M_1\le S_2\le M_2$. For the $q$-shifted factorial $(q;q)_r$, with sums over integer tuples satisfying the displayed constraints, consider the two multiple basic hypergeometric expressions

$$\sum_{k_1+\dots+k_n=S_1}(-1)^{(n-1)S_1}q^{\frac{n(n+m)}2\sum_{i=1}^n k_i^2+m\sum_{i=1}^n i k_i-m\binom{S_1+1}{2}-nS_1(S_1+m)/2} \prod_{1\le i<j\le n}(1-q^{n k_j-n k_i+j-i}) \prod_{i=1}^n\frac{(q;q)_{M_1+M_2+i-1}}{(q;q)_{M_1-S_1+n k_i+i-1}(q;q)_{M_2+S_1-n k_i+n-i}}$$

and

$$\sum_{l_1+\dots+l_m=S_2}(-1)^{(m-1)S_2}q^{\frac{m(m+n)}2\sum_{i=1}^m l_i^2+n\sum_{i=1}^m i l_i-n\binom{S_2+1}{2}-mS_2(S_2+n)/2} \prod_{1\le i<j\le m}(1-q^{m l_j-m l_i+j-i}) \prod_{i=1}^m\frac{(q;q)_{M_1+M_2+i-1}}{(q;q)_{M_1-S_2+m l_i+i-1}(q;q)_{M_2+S_2-m l_i+m-i}}.$$

Warnaar's conjecture. These two expressions are equal. This conjectural identity would extend the hierarchy of transformations between multiple basic hypergeometric series of different dimensions, generalizing Milne's identity and the theorem proved earlier in the paper. The source reports overwhelming computer-experimental evidence, but the conjecture is presented as a proposed generalization and is not established there.