Demonet’s conjecture (E-finite implies brick-finite)

Determine whether E-finiteness—i.e., for every τ-regular irreducible component Z of the representation varieties of A, the generic number of parameters satisfies c(Z)=0—implies brick-finiteness, i.e., that A has only finitely many bricks up to isomorphism.

Background

The authors define E-finiteness via τ-regular components (Plamondon’s framework) and connect it to the behavior of the τ-tilting fan and brick-finiteness. Demonet originally posed a closely related question about rational rays outside the τ-tilting fan; the present conjecture is presented as a reformulation.

Establishing this implication would link homological–geometric finiteness of τ-regular components with the algebraic finiteness of bricks, clarifying the landscape of modern finiteness notions.