Cohen–Macaulayness conjecture for $r$-co-connected complexes

Let $G$ be a finite simple graph. For an integer $r$, let $\Sigma_r(G)$ denote its $r$-co-connected complex, and let $\operatorname{Con}_r(G)$ denote the associated $r$-connected graph.

Cohen–Macaulayness conjecture. The complex $\Sigma_r(G)$ is Cohen–Macaulay if and only if $\operatorname{Con}_r(G)$ is co-chordal.

This conjecture proposes a characterization of Cohen–Macaulayness for $r$-co-connected complexes in terms of the co-chordality of the associated $r$-connected graph. The surrounding discussion gives examples and structural results for several graph classes, but does not establish the claimed equivalence in general.