Equivalence to brick-finiteness for tame algebras

Ascertain whether, for every tame finite-dimensional algebra A over an algebraically closed field, brick-finiteness is equivalent to each of the following five conditions: (i) A is brick-discrete; (ii) A is E-finite; (iii) A is stably-discrete; (iv) the Auslander–Reiten quiver Γ_A has no generalized standard components; and (v) Γ_A has no infinite family of Hom-orthogonal tubes whose quasi-simple modules all have the same dimension.

Background

The authors prove an equivalence of five conditions for tame algebras (brick-discrete, E-finite, stably-discrete, absence of generalized standard components, and absence of certain Hom-orthogonal tube families). They note that the 2nd bBT conjecture remains open for arbitrary tame algebras and highlight an additional unresolved equivalence to brick-finiteness.

Settling this would complete the equivalence diagram for tame algebras, tying geometric, homological, and Auslander–Reiten–theoretic properties directly to brick-finiteness.