Shellability of dominant-composition posets with repeated parts

Prove that, for every partition with repeated elements, the poset of dominant compositions associated with an arborescent poset and its consistent anti-linearization is shellable.

Background

Section 2 studies the Bruhat order on compositions that are dominant with respect to an arborescent poset equipped with a consistent anti-linearization. For regular partitions, the authors prove that the resulting dominant-composition poset is bounded, graded, subthin, and EL-shellable.

When the partition has repeated elements, the corresponding parabolic Bruhat setting can fail to be graded, as illustrated by Example 2.42. The authors state shellability as a conjectural property that persists despite this failure of gradedness. This conjecture is part of the combinatorial foundation used later to analyze Möbius functions and coefficients in the Cauchy identities for staircase matrices.