The stratified cohomological Hall algebra amalgamation conjecture

Let $Z=Z_1\cup\cdots\cup Z_s$ be a union of curves, let $\langle Z\rangle$ be the relevant semigroup of classes, and let $\overline{\mathfrak a}_\gamma$ be the amalgamation map in class $\gamma$. Let $\tau_\gamma$ be the map from the tensor product over equivariant cohomology to the iterated tensor product over the cohomological Hall algebras of pairwise intersections, and let $\ker(\tau_\gamma)\to\ker(\mathfrak a_\gamma)$ be the induced kernel map. The stratified cohomological Hall algebra amalgamation conjecture. For every $\gamma\in\langle Z\rangle$, the map $\overline{\mathfrak a}_\gamma$ is topologically surjective, and the induced map

$$\ker(\tau_\gamma)\longrightarrow\ker(\mathfrak a_\gamma)$$

is also topologically surjective. This is the multi-component version of the amalgamation prediction for cohomological Hall algebras. The first assertion is stated as proved conditionally in Theorem stated in the supplied status evidence; the context does not establish the second assertion, so the combined conjectural package is recorded as solved only to the extent supported by that evidence.