The one-away multiline-queue enumeration conjecture

Let $N\ge n>k\ge 1$ and let $0\le b_1<\dots<b_n\le N-1$. For the permutation $s_kw_0$ obtained by applying the simple reflection $s_k$ to the longest permutation $w_0$, let $A_k$ be the matrix whose entries are $\binom{b_i+j-1}{j-1}$ in rows $1\le i\le n-k$ and $\binom{b_i+j-2}{j-2}$ in rows $n-k<i\le n$. The one-away MLQ enumeration conjecture. The number of multiline queues with bottom row $s_kw_0$ is

$$G_{s_kw_0}(b_1,\dots,b_n;N)=\binom{N}{k}\det A_k-G_{w_0}.$$

This conjecture gives an explicit determinant formula for multiline queues whose bottom row differs from the longest permutation by one simple reflection. Its continuous analogue is stated later in the paper, while the general multi-reflection case is formulated separately.