The conjecture that an off-diagonal common pair has a common component

Let $H_1$ and $H_2$ be graphs, and let $p\in(0,1)$. A pair $(H_1,H_2)$ is $(p,1-p)$-common when it satisfies the off-diagonal common-pair inequality for every complementary pair of graphons. Common-component conjecture. If there exists $p\in(0,1)$ such that $(H_1,H_2)$ is $(p,1-p)$-common, then at least one of $H_1$ or $H_2$ is common. The theorem exhibiting a pair with an uncommon component shows that both components need not be common, while this weaker assertion remains open.