Hopkins's characteristic-class conjecture for block upper-triangular groups

Let $P_{a|b}\subset \operatorname{GL}_{a+b}(\mathbb{C})$ be the subgroup of block upper-triangular matrices with diagonal blocks of sizes $a$ and $b$. The inclusion of the diagonal block subgroup induces a map

$$\mathrm{H}^{2n}(P_{a|b};\mathbb{Z}_{\mathbb{C}}(n))\longrightarrow \mathrm{H}^{2n}(\operatorname{GL}_a(\mathbb{C})\times\operatorname{GL}_b(\mathbb{C});\mathbb{Z}_{\mathbb{C}}(n)).$$

Hopkins's block upper-triangular characteristic-class conjecture. This induced map is an isomorphism. The statement appears among results attributed to ideas of Bott, but the source provides no resolution or additional context establishing its status.