Williamson–Riche character formula conjecture for tilting modules

Let $\mathbf{G}$ be a reductive algebraic group whose derived subgroup is simply connected, let $\mathbf{B}\subset\mathbf{G}$ be a Borel subgroup, and let $\mathbf{T}\subset\mathbf{B}$ be a maximal torus. Assume that $p>h$, where $h$ is the Coxeter number of $\mathbf{G}$. Let $W_{\mathrm{ext}}$ be the extended affine Weyl group, let ${}^0W_{\mathrm{ext}}$ denote the relevant set of minimal representatives, let $\mathsf{T}_w$ be the corresponding tilting object, let $\mathsf{N}_y$ be the corresponding antispherical object, and let $(\mathsf{T}_w:\mathsf{N}_y)$ be the multiplicity of $\mathsf{N}_y$ in a costandard filtration of $\mathsf{T}_w$. Finally, write ${}^p n_{y,w}(1)$ for the value at $1$ of the $p$-canonical antispherical coefficient. Williamson–Riche character formula conjecture. For all $y,w\in{}^0W_{\mathrm{ext}}$,

$$(\mathsf{T}_w:\mathsf{N}_y)={}^p n_{y,w}(1).$$

This character formula was conjectured by G. Williamson and the second author and relates multiplicities in costandard filtrations of tilting modules to the $p$-canonical basis. The supplied text does not state whether the conjecture has been proved or disproved.