Disjoint-union equality

Prove or disprove that, for all graphs $H_1$ and $H_2$, the Ramsey function of their disjoint union satisfies $#2{^n}{H_1\sqcup H_2}=\min\{#2{^n}{H_1},#2{^n}{H_2}\}$ for all sufficiently large dimensions $n$.

Background

The paper notes the immediate upper bound obtained from the fact that each component is a subgraph of the disjoint union, but does not know whether equality always holds asymptotically.

References

Is it true that $#2{n}{H_1 \sqcup H_2} = \min {#2{n}{H_1}, #2{n}{H_2}}$ at least for large enough $n$?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers  (2512.15516 - Axenovich et al., 17 Dec 2025) in Question, Section 6.2 (Growing dimension)