Frobenius–Perron dimension bounds for Brauer tree algebras

Determine whether, for every multiplicity-free Brauer tree algebra associated with a tree having n edges, the Frobenius–Perron dimensions satisfy the inequalities FPdim(A_star) ≤ FPdim(A_G) ≤ FPdim(A_line), where A_star and A_line are the multiplicity-free Brauer star and Brauer line algebras, respectively, each with n non-isomorphic simple modules.

Background

The paper establishes that the Frobenius–Perron dimension of every multiplicity-free Brauer tree algebra agrees with the Frobenius–Perron dimension of its finite T-tilting lattice. It then singles out two extremal-looking tree shapes: the star and the line.

For a Brauer star algebra, the authors state that the Frobenius–Perron dimension is exactly one. For a Brauer line algebra with n non-isomorphic simple modules, they state that the Frobenius–Perron dimension is at least 2 cos(π/(n+1)). Question 3.8 asks whether these two algebras bound the Frobenius–Perron dimension of every multiplicity-free Brauer tree algebra with the same number of edges.