Totaro's cycle-map conjecture for Brown–Peterson cohomology of classifying spaces

Let $G$ be a complex linear algebraic group, let $CH^i(BG)_{(p)}$ be the $p$-localized Chow group of its classifying space, and let $BP^*(BG)$ denote its Brown–Peterson cohomology. Assume that $BP^*(BG)$ has no nonzero odd-degree element. Totaro's conjecture. The cycle map

$$CH^i(BG)_{(p)} \to \left(\mathbb{Z}_{(p)}\otimes_{BP^*}BP^*(BG)\right)^{2i}$$

is an isomorphism. The conjecture links the Chow ring of an algebraic-group classifying space with Brown–Peterson cohomology. The source does not provide evidence of a resolution, so its status is left open.