Elmer's separating-set conjecture for quiver representations

Let $Q$ be a quiver, and for each dimension vector $\alpha$ let $\mathcal{V}_{\alpha}$ be its representation variety, with $G=\prod_i \operatorname{SL}_{\alpha(x_i)}(\mathbb{C})$ acting on it. A polynomial separating set is a separating algebra that is a polynomial ring, and a hypersurface separating set is a separating algebra that is a hypersurface. Elmer's conjecture. The following are equivalent:

1. $Q$ has finite or tame representation type.
2. $\mathbb{C}[\mathcal{V}_{\alpha}]^G$ contains a polynomial or hypersurface separating set, for each dimension vector $\alpha$.

This conjecture generalizes the Skowronski–Weyman characterization of finite or tame quivers via polynomial or hypersurface invariant algebras, replacing the full invariant algebra by a separating set. The source presents it as an open conjecture; no resolution is supplied here.