Lusztig's Fourier-matrix conjecture for non-crystallographic Coxeter groups

Let $W$ be a non-crystallographic finite Coxeter group, let $c$ be a two-sided cell in $W$, and let $\mathcal{U}_W^c=\mathcal{Z}(\mathcal{H}_W^c)$ be the Drinfeld center of the asymptotic Hecke category associated with $c$. Denote its $S$-matrix by $S(\mathcal{U}_W^c)$ and let $F_W^c$ be the Fourier matrix associated with $W$ and $c$. Lusztig's conjecture. The $S$-matrix of $\mathcal{U}_W^c$ is equal to $F_W^c$. In particular, the number of simple objects of $\mathcal{U}_W^c$ is equal to the number of unipotent characters supported in $c$. This extends the expected relationship between centers of asymptotic Hecke categories and unipotent character data to non-crystallographic types; the modular tensor category assertion for these centers was proved by Elias and Williamson, while this Fourier-matrix identification remains the stated conjecture.