Remarks on $\tau$-tilted versions of the second Brauer-Thrall Conjecture

In this short note, we state a stable and a $\tau$-reduced version of the second Brauer-Thrall Conjecture. The former is a slight strengthening of a brick version of the second Brauer-Thrall Conjecture raised by Mousavand and Schroll-Treffinger-Valdivieso. The latter is stated in terms of Geiss-Leclerc-Schr\"oer's generically $\tau$-reduced components and provides a geometric interpretation of a question of Demonet. It follows that the stable second Brauer-Thrall Conjecture implies our $\tau$-reduced second Brauer-Thrall Conjecture. Finally, we prove the reversed implication for the class of E-tame algebras recently introduced by Asai-Iyama.