Let $G$ be a minimal $2$-colour Ramsey-graph for $H$.

Must there exist a minimal $3$-colour Ramsey-graph $F$ for $H$ with $G\subset F$?

I am wondering if anything is known about this, particularly in the case $H=K_n$, at least for $n=3$.

(By G being an $r$-Ramsey-graph for $H$ I mean the property that every colouring of the edges of $G$ with $r$ colours admits a monochromatic copy of $H$. By minimal I mean minimal wrt. the subgraph relation.)