Casselman's regularity conjecture for cell languages

Let $W$ be a Coxeter group with generating set $S$, and let $C\subset W$ be a cell. Write $\mathtt{Red}(C)$ for the language of reduced expressions of elements of $C$. Casselman's regularity conjecture. For any Coxeter group $W$ and any cell $C\subset W$, there exists a finite state automaton accepting the language $\mathtt{Red}(C)$. That is, the language $\mathtt{Red}(C)$ is regular. This conjecture asks whether the generally infinite combinatorial data of every cell can be encoded by a finite-state automaton. The source gives no resolution in the stated generality; in the paper, regularity is established for cells in the affine Weyl groups studied there.