Asymmetric Ramsey-density conjecture for graphs

Let $H$ and $H'$ be graphs with $m_2(H)\geq m_2(H')$. Asymmetric Ramsey-density conjecture. If neither $H$ nor $H'$ is a matching and at least one of $H$ or $H'$ contains a cycle, then

$$m(F)>m_2(H,H')$$

for every Ramsey graph $F\in R(H,H')$. This conjecture would, together with asymmetric random-Ramsey results, imply the expected Ramsey-infinite classification for unordered graph pairs containing cycles; the source gives no resolution.