Katsylo's birationality conjecture for generically free representations

Let $G$ be an algebraic group, and let $V$ and $W$ be generically free linear representations of $G$. Katsylo's conjecture. If

$$\dim(V)=\dim(W),$$

then $V$ and $W$ are birationally isomorphic as $G$-varieties. This conjecture concerns the birational classification of generically free linear representations; stable birational isomorphism follows from the no-name lemma, while the stronger birational assertion is the proposed classification problem.