Dual EL-shellability conjecture for Coxeter-type admissible sets

Let $\operatorname{Cox}(\mu)^K$ be the subset of $\operatorname{Adm}(\mu)$ consisting of the spherical Coxeter elements associated with the parahoric subgroup $K$. Adjoin a maximum element to obtain

$$\widehat{\operatorname{Cox}(\mu)^K}=\operatorname{Cox}(\mu)^K\sqcup\{\hat 1\}.$$

Dual EL-shellability conjecture. The poset $\widehat{\operatorname{Cox}(\mu)^K}$ is dual EL-shellable.

This conjecture concerns the combinatorics governing EKOR strata in basic loci of Coxeter type and is motivated by empirical observations. Its resolution is not given in the source.