CKW conjecture (stably-discrete iff brick-discrete)

Prove that for every finite-dimensional algebra A over an algebraically closed field, A is stably-discrete—meaning that for every irreducible component Z of rep(A,\mathbf{d}) and every weight \theta in K_0(proj A), the moduli space \mathcal{M}^{\theta-ss}(Z) is zero-dimensional—if and only if A is brick-discrete—meaning every brick has an open orbit in its irreducible component.

Background

The paper introduces the stably-discrete property via King’s moduli of semistable representations and relates it to brick-discreteness (openness of brick orbits). For tame algebras, the authors prove the equivalence (Theorem [CKW]), and then state the general (all algebras) version as an open conjecture.

Resolving this conjecture would unify geometric stability behavior (semi-invariant theory/moduli) with orbit geometry of bricks across all finite-dimensional algebras.