The Gallai-Ramsey reduction conjecture for rainbow $P_{5}$

Let $P_{5}$ be the path on five vertices, let $H$ be a graph with no isolated vertices, and let $k$ be a positive integer. The Gallai-Ramsey number $gr_k(P_{5}:H)$ is the minimum integer $n$ such that every coloring of $K_n$ using at most $k$ colors contains either a rainbow copy of $P_{5}$ or a monochromatic copy of $H$. Let $R_3(H)$ denote the minimum integer $n$ such that every coloring of $K_n$ using at most three colors contains a monochromatic copy of $H$.

Gallai-Ramsey reduction conjecture. For any graph $H$ with no isolated vertices, we have

$$gr_k(P_{5}:H)=R_3(H).$$

The conjecture asserts that the Gallai-Ramsey problem for avoiding a rainbow $P_{5}$ reduces to the three-color Ramsey number. The paper provides several results in support of this claim, but no resolution is supplied here.