The finite maximal-path conjecture for coclass graphs of nilpotent semigroups

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. The graph $\mathcal G_{r,K}$ has rooted-tree components, and a maximal infinite path is an infinite path starting at the root of its rooted tree. Finite maximal-path conjecture. The graph $\mathcal G_{r,K}$ has only finitely many maximal infinite paths. Their number depends on $r$ but not on $K$. Equivalently, $\mathcal G_{r,K}$ consists of finitely many maximal coclass trees and finitely many other vertices. This is the first general coclass conjecture proposed for these semigroup graphs; its resolution is not supplied here.