Rational-singularities conjecture for mixed-characteristic local models

Let $\underline{\mathcal{G}}$ be the relevant parahoric group scheme and let $\mu$ be the specified cocharacter datum, and write $\widetilde{M}_{\underline{\mathcal{G}},\mu}$ for the corresponding local model. Rational-singularities conjecture. The local model $\widetilde{M}_{\underline{\mathcal{G}},\mu}$ has rational singularities. Rational singularities are known in the equicharacteristic case and at special level in mixed characteristic, which motivates this conjecture for the general mixed-characteristic local model; the supplied source gives no resolution.