Geiss–Leclerc–Schröer conjecture on square-irreducibility

Let $\mathfrak{m}$ be a multisegment, let $V$ be the graded vector space determined by its supercuspidal support, and let $\mathfrak{C}_{\mathfrak{m}}$ be the corresponding irreducible component of the commuting variety. Write $GLS(\mathfrak{m})$ for the condition that $\mathfrak{C}_{\mathfrak{m}}$ admits an open, equivalently dense, $\mathbb{G}(V)$-orbit. Geiss–Leclerc–Schröer conjecture. For any multisegment $\mathfrak{m}$, $Z(\mathfrak{m})$ is $\square$-irreducible if and only if $GLS(\mathfrak{m})$ holds. The conjecture is known for regular multisegments and has been verified computationally for multisegments consisting of at most six segments, but remains open in general.