The homological-dimension-one conjecture for self-dual representations

Let $G$ be a connected reductive group, let ${\mathsf V}$ be a self-dual $G$-module, and suppose that the defect of the null-cone satisfies

$${\mathrm{def\,}}{\mathfrak N}_G({\mathsf V})=1.$$

Homological-dimension-one conjecture. The quotient ${\mathsf V}/\!\!/G$ is either an affine space or a hypersurface; equivalently,

$${\mathrm{hd\,}}\,{\mathsf V}/\!\!/G\leqslant 1.$$

This is a conjecture about the invariant-theoretic quotient of a non-equidimensional representation, extending the preceding defect-zero conjecture. The supplied context does not state which cases are known or whether this conjecture has been resolved.