  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.
