Tezuka's cohomology conjecture for loop groups and finite Chevalley groups

Let $G$ be the group considered in the paper, let $BG$ be its classifying space, and let $\mathcal{L}BG$ denote its free loop space. Write $G(\mathbb{F}_q)$ for the corresponding finite Chevalley group over the finite field with $q$ elements. If $\ell$ is an odd prime dividing $q-1$, or if $\ell=2$ and $4$ divides $q-1$, then Tezuka's conjecture. There is a ring isomorphism

$$H^{*}(\mathcal{L}BG;\mathbb{Z}/\ell)\cong H^{*}(BG(\mathbb{F}_q);\mathbb{Z}/\ell).$$

The conjecture compares the mod-$\ell$ cohomology of the loop-group classifying space with that of the associated finite Chevalley group, despite the two objects being different both as topological spaces and as groups. Its status is not resolved in the supplied source.