The separating-set characterization of tame quiver representation type

Let $Q$ be a quiver. For a dimension vector $\alpha$, let $\mathcal{V}_{\alpha}$ be the representation space of $Q$ of dimension vector $\alpha$, and let $G=\prod_{i=1}^k\operatorname{SL}_{\alpha(x_i)}(\mathbb{C})$ act on it. A separating set is a collection of semi-invariants that distinguishes any two points distinguished by some semi-invariant.

Separating-set conjecture. The following are equivalent:

$$Q\text{ has tame representation type};$$ $$\mathbb{C}[\mathcal{V}_{\alpha}]^G\text{ contains a polynomial or hypersurface separating set for every dimension vector }\alpha.$$

This conjecture generalizes the Skowroński–Weyman characterization, which replaces “contains a polynomial or hypersurface separating set” by “is a polynomial ring or hypersurface.” The finite-representation-type case is consistent with the conjecture because the semi-invariant algebras are polynomial for all dimension vectors, while the equivalence for tame quivers remains open here.