Mod-p cohomological rigidity conjecture for classifying spaces of compact Lie groups

Let $G$ be a compact connected Lie group, let $X$ be a $p$-complete space, and let $\mathcal{A}_p$ denote the mod-$p$ Steenrod algebra. Suppose that

$$H^*(X;\mathbb F_p)\cong H^*(BG;\mathbb F_p)$$

as algebras over $\mathcal{A}_p$. Cohomological rigidity conjecture. Then

$$X\simeq BG_p^{\wedge}.$$

The claim asks whether the mod-$p$ Steenrod-algebra cohomology of a compact Lie group's classifying space determines its $p$-completion among $p$-complete spaces. The paper states this as a conjecture and proves the corresponding rigidity result for $G=PU(p)$ when $p>3$, extending earlier results for $p=2,3$; no general resolution is supplied.