Closure of (strongly) sat-sharp graphs under disjoint union

Let $H_1$ and $H_2$ be graphs that are (strongly) sat-sharp, and let $H_1+H_2$ denote their disjoint union. Disjoint-union conjecture. If $H_1$ and $H_2$ are (strongly) sat-sharp graphs, then $H_1+H_2$ is (strongly) sat-sharp; equivalently, the class of (strongly) sat-sharp graphs is closed under taking disjoint unions. The preceding results establish this behavior for certain classes, including disjoint unions of cliques, but the general closure assertion is presented as plausible and remains open.