The finite-group or monomiality conjecture for group-type BVSs

Let $(V,c)$ be a group-type BVS, and let $G=\langle g_i\rangle$ be the group generated by its associated operators. Assume that $c$ is unitary and has finite order. Finite-group or monomiality conjecture. Either $G$ is a finite group, or there is a basis of $V$ with respect to which $c$ is monomial. The conjecture is a stricter version of the paper's main conjecture in the setting of unitary, finite-order group-type BVSs; the supplied text does not state whether it has been proved or refuted.