Koszul cohomology criterion for formality of flag moment-angle complexes

Let $\mathcal{K}$ be a flag simplicial complex, and consider the grading on $H^*(\mathcal{Z}_\mathcal{K};\mathbb{Q})$ specified in the source's preceding theorem. Koszul formality conjecture. If $H^*(\mathcal{Z}_\mathcal{K};\mathbb{Q})$ is Koszul with respect to that grading, then $\mathcal{Z}_\mathcal{K}$ is formal over $\mathbb{Q}$. This is the converse direction to the preceding formality criterion, which proves the Koszul property under rational formality. The supplied text does not provide a resolution.