Remarks on τ$\tau$‐tilted versions of the second Brauer–Thrall conjecture

Abstract In this short note, we state a stable and a ‐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–Schröer's generically ‐reduced components and provides a geometric interpretation of a question of Demonet. It follows that the stable second Brauer–Thrall conjecture implies our ‐reduced second Brauer–Thrall conjecture. Finally, we prove the reversed implication for the class of ‐tame algebras recently introduced by Asai–Iyama.