Fox–Grinshpun–Pach conjecture for Gallai–Ramsey numbers of triangles and cliques

Let $k\geq 3$ and $p\geq 3$ be integers. Write $r(K_p,K_p)$ for the two-color Ramsey number of the complete graph $K_p$, and let $\operatorname{gr}_k(K_3:K_p)$ denote the Gallai–Ramsey number for a rainbow triangle and a monochromatic $K_p$. (The displayed formula in the source uses $n$ for the parity condition, although the quantified parameter is $k$.)

Fox–Grinshpun–Pach conjecture.

$$\operatorname{gr}_k(K_3:K_p)=\left\{\begin{array}{ll}(r(K_p,K_p)-1)^{k/2}+1, & \text{if }k\text{ is even},\\(p-1)(r(K_p,K_p)-1)^{(k-1)/2}+1, & \text{if }k\text{ is odd}.\end{array}\right.$$

This conjecture gives an exact formula for the Gallai–Ramsey number of a rainbow triangle versus a monochromatic complete graph, refining the preceding asymptotic result. The supplied material gives no evidence of a resolution, so its status remains open.