Primitively generated loop homology conjecture for Davis–Januszkiewicz spaces

Let $\mathcal{K}$ be a simplicial complex and let $\mathbf{k}$ be a ring such that $H_*(\Omega\mathrm{DJ}(\mathcal{K});\mathbf{k})$ is a free $\mathbf{k}$-module. Recall that a Hopf algebra is primitively generated if it is multiplicatively generated by its primitive elements, where an element $x$ is primitive when $\Delta x=x\otimes 1+1\otimes x$. Primitively generated loop homology conjecture. The Hopf algebra $H_*(\Omega\mathrm{DJ}(\mathcal{K});\mathbf{k})$ is primitively generated for every such $\mathcal{K}$ and $\mathbf{k}$. This would describe the loop-homology Hopf algebra using its primitive elements; the supplied text does not state whether the claim has been resolved.