Conjecture on Gallai–Ramsey numbers of even wheels

Let $W_{2n}$ denote the wheel graph with an even cycle of length $2n$, let $GR_k(H)$ denote the $k$-color Gallai–Ramsey number of a graph $H$, and let $R_2(W_{2n})$ denote the two-color Ramsey number of $W_{2n}$. Even-wheel Gallai–Ramsey conjecture. For all $k\ge2$ and $n\ge2$,

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

The formula is motivated by the exact computation of $GR_k(W_4)$ for all $k\ge2$, together with a lower bound for general even wheels. Its validity for all even wheels and all $k\ge2$ remains open.