Note: This problem will be also be explained on my research talk on Wednesday.
An algebra is said to be g-tame if the union of all the cones spanned by a τ-tilting pairs is dense in R^n.
The statement of the second τ-Brauer-Thrall conjecture states that if an algebra is τ-tilting finite, then there exists an infinite family of bricks (i.e. modules whose endomorphism algebra is a division ring) {B_j : j \in J} such that dim_k B_j = d for all j \in J.
We want to show the second τ-Brauer-Thrall conjecture for g-tame algebras. In order to do this, we will first see that every τ-tilting infinite algebra has an infinite chain of torsion classes in their lattice of torsion classes. Then we will realise this infinite chain of torsion classes as an infinite set of distinct points in the wall-and-chamber of the algebra in such a way that this sequence of points has a limit. We then will study the category of stable modules in this limit point. If we are lucky, we can find an invite family of stable objects at this point. Since every stable module is necessarily a brick, then we would have proven the second τ-Brauer-Thrall conjecture for g-tame algebras.
The second τ-Brauer-Thrall conjecture sounds interesting, and seems related to the following conjecture:
If any θ \in K_0(proj A) has a (not necessarily basic) 2-term presilting complex U such that θ=[U], then A is τ-tilting finite.
Note that we only consider the elements in the ordinary Grothendieck group, not the real Grothendieck group.
We don't know whether this conjecture holds true or not even for representation tame algebras.