The multi-away multiline-queue enumeration conjecture

Let $N\ge n$, let $\mathbf{k}=(k_1\ge\cdots\ge k_r\ge 1)$ be a partition, and let $\mathbf{k}'$ be its conjugate. For $S\subseteq [r]$, let $\mathbf{k}(S)$ be the partition consisting of the parts $k_i$ with $i\in S$, and let $\mathbf{k}(S)'$ be its conjugate. Define $A_S$ to have entries

$$\binom{b_i+j-1-k_{n+1-i}(S)'}{j-1-k_{n+1-i}(S)'}.$$

Here $k_i(S)'=0$ if $i>k_{\min S}$. Assume $n>k_1>k_2+1>k_3+2>\dots>k_r+r-1>r-1$ and $0\le b_1<\dots<b_n\le N-1$. The multi-away MLQ enumeration conjecture. The number of multiline queues is

$$G_{s_{k_1}\dots s_{k_r}w_0}(b_1,\dots,b_n;N)=\sum_{S\subseteq [r]}(-1)^{|S|}\prod_{i\in S}\binom{N}{k_i}\det A_S.$$

This generalizes the one-away formula to commuting simple reflections. The case $r=1$ specializes to the one-away conjecture, and the paper does not provide a proof of the general statement.