Cohomology comparison conjecture for saturable pro-$p$ groups

Let $G$ be a saturable pro-$p$ group with associated $\mathbb{Z}_p$-Lie algebra $\mathfrak{g}$. Here $\operatorname{H}^{*}(G;\mathbb{F}_p)$ and $\operatorname{H}^{*}(\mathfrak{g};\mathbb{F}_p)$ denote the cohomology algebras of $G$ and $\mathfrak{g}$, respectively, and $\mathfrak{g}/p\mathfrak{g}$ is the reduction of the Lie algebra modulo $p$. Cohomology comparison conjecture. There is an isomorphism of $\mathbb{F}_p$-algebras

$$\operatorname{H}^{*}(G;\mathbb{F}_p)\cong \operatorname{H}^{*}(\mathfrak{g};\mathbb{F}_p)\cong \operatorname{H}^{*}(\mathfrak{g}/p\mathfrak{g};\mathbb{F}_p).$$

This extends Lazard's cohomology comparison from uniform or equi-$p$-valued groups to the broader class of saturable pro-$p$ groups; the conjecture is presented as an extension and its resolution is not supplied here.