Kanade–Russell conjecture for the complete family of Andrews–Schilling–Warnaar identities

Let $a,b,k$ be integers such that $0\leqslant a,b\leqslant k$, let $\tau\in\{-1,0,1\}$, and set

$$K:=3k+\tau+3.$$

For weakly decreasing nonnegative integer sequences $\lambda_1\geqslant\cdots\geqslant\lambda_k\geqslant0$ and $\mu_1\geqslant\cdots\geqslant\mu_k\geqslant0$, with $q^{\lambda_0}=q^{\mu_0}:=0$, and with $g_{\lambda_k,\mu_k;\tau}(q)$ as in the Andrews–Schilling–Warnaar identities, Kanade–Russell conjecture.

$$\begin{aligned} &\sum_{\substack{\lambda_1\geqslant\cdots\geqslant\lambda_k\geqslant0\\ \mu_1\geqslant\cdots\geqslant\mu_k\geqslant0}} \frac{1-q^{\lambda_a+\mu_b+1}}{1-q} \frac{q^{\sum_{i=1}^k(\lambda_i^2-\lambda_i\mu_i+\mu_i^2)+\sum_{i=a+1}^k\lambda_i+\sum_{i=b+1}^k\mu_i}} {\prod_{i=1}^{k-1}(q;q)_{\lambda_i-\lambda_{i+1}}(q;q)_{\mu_i-\mu_{i+1}}} g_{\lambda_k,\mu_k;\tau}(q)\\ &\qquad=\frac{(q^K;q^K)_\infty^2}{(q;q)_\infty^3}\,\theta(q^{a+1},q^{b+1},q^{a+b+2};q^K). \end{aligned}$$

This conjecture completes the known ASW identities, providing the expected multisum and product identity for every pair of parameters $a,b$ with $0\leqslant a,b\leqslant k$. It has interpretations in terms of principally specialised characters of $\mathrm{A}_2^{(1)}$ and generating functions for cylindric partitions, but its general status is not established.