Su and Liu's Ramsey-full equivalence conjecture

Let $H$ be a graph with no isolated vertex. A graph is Ramsey-full if, writing $R(H,H)=N$, there exists a red-blue edge-coloring of $K_N-e$ containing neither a red copy nor a blue copy of $H$. It is Gallai-Ramsey-full if, writing $GR_k(H)=N$, there exists a Gallai $k$-edge-coloring of $K_N-e$ using colors from $[k]$ with no monochromatic copy of $H$ in any color.

Su and Liu's conjecture. $H$ is Ramsey-full if and only if $H$ is Gallai-Ramsey-full.

Su and Liu observed that complete graphs and cycles of length four have both properties. The conjecture is presented here as open; this paper gives two classes of graphs that are Ramsey-full but not Gallai-Ramsey-full, disproving it.