Conjecture on Coxeter-type affine Deligne-Lusztig stratifications

Let $G$ be a reductive group, let $mu$ be a Hodge cocharacter, and let $K$ be a parahoric subgroup. Assume that $(G,\mu,K)$ is of Coxeter type, and let the corresponding basic affine Deligne-Lusztig variety be equipped with its Ekedahl–Oort stratification and the decomposition determined by the functions $f_g$. Coxeter-type stratification conjecture. The paving of the corresponding basic affine Deligne-Lusztig variety by classical Deligne–Lusztig varieties induced by the Ekedahl–Oort stratification coincides with the decomposition according to the functions $f_g$. This predicts that the two natural stratifications agree in every Coxeter-type case.