Inclusion conjecture for ordered matroid Hopf monoids and shellable complexes

Let $\textbf{\textsc{omat}}_+$ be the class of ordered matroid complexes with the indicated extension, and let $\textbf{\textsc{sls}}$ be the class of shellable prefix-pure complexes. Inclusion conjecture.

$$\textbf{\textsc{omat}}_+\subseteq\textbf{\textsc{sls}}.$$

This conjecture concerns shellability of the complexes underlying the ordered matroid Hopf monoid. The supplied status evidence says that the assertion was proved: every generic real-valued function on a matroid ground set, extended linearly to bases, yields a shelling order on the independence complex.