Type B Hurwitz orbit cardinality conjecture for grouped Coxeter factorizations

Fix simple reflections $s_0,s_1,\ldots,s_{kn-1}$ in the hyperoctahedral group of type $B_{kn}$, with $(s_0s_1)^4=1$, and let $\mathbf{t}$ be the grouped factorization

$$\mathbf{t}=(s_0\cdots s_{k-1})\cdot(s_k\cdots s_{2k-1})\cdots(s_{kn-k}\cdots s_{kn-1}).$$

The Hurwitz orbit of $\mathbf{t}$ contains $k^{n-1}n^n$ elements. This is proposed as a type $B$ analogue of the enumerative results for the grouped factorizations studied in the paper; its status is open.