Real Bredon cohomology comparison conjecture for algebraic groups

Let $G$ be an algebraic group over $\mathbb R$, let

$$\Gamma=\operatorname{Gal}(\mathbb C/\mathbb R),$$

and let $p$ be a prime number. Regard $G(\mathbb C)$ as a $\Gamma$-space and let $G(\mathbb C)^{\operatorname{top}}$ denote its topological realization, with the induced $\Gamma$-action. Real Bredon cohomology comparison conjecture. The identity map

$$G(\mathbb C)\longrightarrow G(\mathbb C)^{\operatorname{top}}$$

should induce an isomorphism

$$H^\bullet_\Gamma(BG(\mathbb C)^{\operatorname{top}};\underline{\mathbb Z/p})\longrightarrow H^\bullet_\Gamma(BG(\mathbb C);\underline{\mathbb Z/p}).$$

This is proposed as a comparison between the algebraic and topological models using equivariant Bredon cohomology. The source says that the preceding fixed-point criterion suggests this statement, but gives no resolution status.