Klein–Elias–Hogancamp conjecture for Soergel bimodules

Let $(W,S)$ be a Coxeter system, let $\mathcal{H}$ be a two-sided cell, and let $d$ be a Duflo involution in $\mathcal{H}$ with $\mathtt{a}(d)=a$. Let $\widetilde{\mathrm{C}}_d$ be the associated $1$-morphism in the cell quotient $\mathscr{S}_\mathcal{H}$. Klein–Elias–Hogancamp conjecture. The $1$-morphism $\widetilde{\mathrm{C}}_d$ is a graded Frobenius algebra object in $\mathscr{S}_\mathcal{H}$, with product, coproduct, unit, and counit obtained from compatible choices of the projections, inclusions, and morphisms specified by the decomposition of $\widetilde{\mathrm{C}}_d\widetilde{\mathrm{C}}_d$. This is known for finite Coxeter groups, while it remains open for infinite Coxeter groups when $d$ is not the longest element of a finite parabolic subgroup.