Song, Wei, Zhang, and Zhao's Gallai-Ramsey conjecture for even wheels

Let $W_m$ be the wheel obtained by joining a central vertex to every vertex of the cycle $C_m$, and let $R_2(W_m)$ be its two-color Ramsey number.

Song, Wei, Zhang, and Zhao's conjecture. For all $k\ge2$ and even $m\ge4$,

$$GR_k(W_m)= \begin{cases} (R_2(W_m)-1)\cdot5^{(k-2)/2}+1, & \text{if } k \text{ is even};\\ 2(R_2(W_m)-1)\cdot5^{(k-3)/2}+1, & \text{if } k \text{ is odd}. \end{cases}$$

The supplied text presents this as the conjecture for general even-order wheels and does not state its resolution.