Geck–Halls' conjecture on left cells and generalized tilde-tau invariants

Let $W_f$ be a finite Coxeter group, and let $x,y\in W_f$. Say that $x$ and $y$ have the same generalized $\widetilde{\tau}$-invariant when they are equivalent under all the relations $\approx_n'$ defined by $x\approx_0' y$ if their right descent sets agree, and by requiring compatibility with every applicable right star operation at each successive stage. Geck–Halls' conjecture. The elements $x$ and $y$ belong to the same Kazhdan–Lusztig left cell if and only if they lie in the same Kazhdan–Lusztig two-sided cell and have the same generalized $\widetilde{\tau}$-invariant. This is a proposed characterization of Kazhdan–Lusztig left cells in finite Coxeter groups; the source presents it as a conjecture attributed to Geck and Halls.