Logarithmic balanced-clique conjecture for graphs with fixed triangle density

Let $G$ be a graph on $n$ vertices with triangle density $\gamma$. For an integer $t$, let $K_{t,t,t}$ denote the balanced complete tripartite graph with three parts of size $t$.

Logarithmic balanced-clique conjecture. There is an absolute constant $c>0$ such that $G$ contains a copy of $K_{t,t,t}$ with

$$t=c\log_{1/\gamma}n.$$

This would match the logarithmic scale suggested by a random graph with edge density $\gamma^{1/3}$. The text presents it as an improvement of Nikiforov's result, and no resolution is given.