The finite-group or monomiality conjecture for group-type braided vector spaces

Let $(V,c)$ be a group-type braided vector space (BVS), meaning a braided vector space of the group-type form developed in the paper. Assume that $c$ is unitary and has finite order, and let $G=\langle g_i\rangle$ be the group generated by the associated elements $g_i$. A basis is monomial for $c$ when the matrix of $c$ in that basis has exactly one nonzero entry in each relevant row and column.

Finite-group or monomiality conjecture. Either $G$ is finite, or there is a basis with respect to which $c$ is monomial.

This is presented as a stricter conjecture in the group-type, finite-order setting, motivated by the broader conjecture on images of unitary braided-vector-space braid representations. The supplied text gives no resolution status.