Zhao and Wei's Gallai-Ramsey conjecture for kipases

For a graph $G$, let $GR_k(G)$ denote its $k$-color Gallai-Ramsey number. A kipas is denoted by $\widehat{K}_m$, and $R_2(\widehat{K}_m)$ is its two-color Ramsey number.

Zhao and Wei's conjecture. For all $k\ge2$ and $m\ge2$,

$$GR_k(\widehat{K}_m)= \begin{cases} (R_2(\widehat{K}_m)-1)\cdot5^{(k-2)/2}+1, & \text{if } k \text{ is even and } m \text{ is odd};\\ R_2(\widehat{K}_m)+\frac m2(5^{k/2}-5), & \text{if } k \text{ is even and } m \text{ is even};\\ \max\{2(R_2(\widehat{K}_m)-1),5m\}\cdot5^{(k-3)/2}+1, & \text{if } k \text{ is odd}. \end{cases}$$

The source introduces this as a conjecture for general kipases and does not state a resolution in the supplied text.