Katsylo's birational equivalence conjecture for generically free modules

Let $G$ be a reductive algebraic group, and let $V$ and $W$ be finite-dimensional algebraic $G$-modules with trivial stabilizers of points in general position.

Katsylo's conjecture. The following properties are equivalent: $\dim V=\dim W$; and there exists a $G$-equivariant birational map

$$V\overset{\simeq}{\dashrightarrow}W.$$

The conjecture concerns birational classification of generically free linear actions. It was disproved by counterexamples found in 2000, including a birational classification of finite-dimensional $G$-modules for diagonalizable $G$.