Filling-surface growth-rate finiteness conjecture

Consider filling surfaces and the asymptotic growth rates of the diameters of their modular flip-graphs. Say that two filling surfaces have the same growth rate when these diameter sequences have the same asymptotic growth rate.

Filling-surface growth-rate finiteness conjecture. The number of topological types of filling surfaces with the same growth rate is finite.

The conjecture asks whether a fixed diameter growth rate can occur for only finitely many topological types of filling surfaces. It is presented as an open classification problem, and no resolution is supplied in the source.