Fox, Grinshpun, and Pach's Gallai-Ramsey conjecture for complete graphs

Let $GR_k(K_p)$ be the smallest positive integer $N$ such that every Gallai coloring of $K_N$ with at most $k$ colors contains a monochromatic copy of $K_p$. Let $R(K_p)$ denote the two-color Ramsey number of $K_p$.

Fox, Grinshpun, and Pach's conjecture. For positive integers $k$ and $p$,

$$GR_k(K_p)= \begin{cases} (R(K_p)-1)^{k/2}+1, & \text{if } k \text{ is even};\\ (p-1)(R(K_p)-1)^{(k-1)/2}+1, & \text{if } k \text{ is odd}. \end{cases}$$

The formula is known for $p=3$ and $p=4$, whereas the precise value remains unresolved for $p=5$. The source notes tension with the conjecture of McKay and Radziszowski, so the general conjecture remains open.