The Golod characterization of co-$H$ moment-angle complexes

Let $K$ be a simplicial complex. The associated moment-angle complex is $(D^2,S^1)^K$, where $(D^2,S^1)$ is the standard polyhedral product pair. A simplicial complex $K$ is Golod if all products and higher Massey products vanish in the appropriate Tor-algebra of its Stanley–Reisner ring.

Golod characterization. The moment-angle complex $(D^2,S^1)^K$ is a co-$H$-space if and only if $K$ is Golod.

This conjecture proposes a topological characterization of Golod complexes through the homotopy type of their moment-angle complexes. Considerable work on the homotopy theory of moment-angle complexes over Golod complexes motivates the statement, but its resolution is not specified here.