The finite-group non-reflection conjecture for the 2-polarization property

Let $V$ be a finite-dimensional complex vector space and let $H\subset\operatorname{GL}(V)$ be a finite group that is not generated by reflections.

Finite-group non-reflection conjecture. Then $V$ does not have the $2$-polarization property.

The conjecture would extend the paper's results on failures of the $2$-polarization property beyond the main theorem for irreducible representations of simple algebraic groups. It is presented as a statement whose establishment would avoid the computation using LiE in the exceptional case $V=\operatorname{S}^3(\mathbb{C}^4)$ and $H=\operatorname{SL}_4$; no resolution is given here.