Inclusion conjecture for broken-circuit, Gale, and quasisymmetric complexes

Let $\textbf{\textsc{bc}}$, $\textbf{\textsc{omat}}^{\operatorname{\mathsf{gale}}}$, $\textbf{\textsc{qe}}$, and $\textbf{\textsc{qi}}$ denote the corresponding classes of prefix-pure complexes. Inclusion conjecture. One has

$$\textbf{\textsc{bc}}\subsetneq\textbf{\textsc{omat}}^{\operatorname{\mathsf{gale}}}\subsetneq\textbf{\textsc{qe}}\cap\textbf{\textsc{qi}}.$$

The first inclusion is close to known results on broken-circuit complexes, and the conjecture holds for shifted complexes. The paper explains that the relevant inclusions into $\textbf{\textsc{qe}}$ and $\textbf{\textsc{qi}}$ are known, while the full strict-inclusion statement is not established.