Exponential multicolor Ramsey bound for balanced complete tripartite graphs

Let $K_{k,k,k}$ denote the complete tripartite graph with three parts of size $k$, and let $r(F;q)$ be the least $n$ such that every $q$-edge-coloring of the complete graph on $n$ vertices contains a monochromatic copy of $F$. Tripartite Ramsey conjecture. For all $q\geq 2$ and all sufficiently large $k$,

$$r(K_{k,k,k};q)\leq q^{Ck},$$

where $C>0$ is an absolute constant. This is the $H=K_3$ case of the anticipated extension of the paper's main blowup result. It would give an exponential-in-$k$ upper bound, and is described as interesting and challenging; no resolution is supplied here.