Enumeration conjecture for nondegenerate hypermatrices avoiding a plane partition

In the format $2\times (k+1)\times k$, let $\Delta=\Delta_{1,(k-1)}$ and let $P=\langle\lambda,\mu\rangle$ be a plane partition with $P\subseteq\Delta$, where $\lambda$ and $\mu$ are integer partitions of format $(k+1)\times k$. A hypermatrix respects $P$ when it satisfies the avoidance condition specified by the paper.

Enumeration conjecture. The number of nondegenerate $2\times (k+1)\times k$ hypermatrices over $\mathbb{F}_q$ which respect $P$ is

$$q^{k^2}(q-1)^{2k}[k+1-\lambda_1]_q[k-\lambda_2]_q\cdots[2-\lambda_k]_q\cdot[k-\mu_1]_q[(k-1)-\mu_2]_q\cdots[1-\mu_k]_q.$$

This conjecture gives the proposed enumeration of nondegenerate hypermatrices avoiding the plane partition $P$; its resolution is not specified in the supplied text.