The cocenter basis conjecture for Hecke algebras of Coxeter groups

Let $(W,S)$ be a Coxeter system, let $R$ be the coefficient ring, and let $\mathcal{H}$ be its Hecke algebra. The cocenter is the quotient $\mathcal{H}/[\mathcal{H},\mathcal{H}]$, where $[\mathcal{H},\mathcal{H}]$ is the $R$-submodule generated by commutators $hh'-h'h$ for $h,h'\in\mathcal{H}$. Write $\operatorname{cl}(W)$ for the set of conjugacy classes of $W$. Cocenter basis conjecture. The cocenter $\mathcal{H}/[\mathcal{H},\mathcal{H}]$ has a basis indexed by the set of conjugacy classes $\operatorname{cl}(W)$. This conjecture extends the known finite and affine cases; the paper reports a partial result but does not establish the full basis statement for any Coxeter group.