Core semisparsity conjecture for quotient representations of polytopes

Let $W$ be a group and let $N$ be a subgroup of $W$. The subgroup $N$ is semisparse in $W$ if it satisfies the semisparsity condition for quotient representations, and let $\operatorname{Core}(W,N)$ denote the core of $N$ in $W$, namely the largest normal subgroup of $W$ contained in $N$.

Core semisparsity conjecture. If $N$ is semisparse in $W$, then $\operatorname{Core}(W,N)$ is also semisparse.

A positive answer would imply that the two open questions described in the paper have negative answers: every finite polytope would have a well-defined quotient construction using the core, and every finite polytope would admit a presentation as a quotient of a finite regular polytope. The source does not report a resolution of this conjecture.