Polynomial Erdős–Hajnal-type dependence for tripartite 3-graphs

Let $H_0$ be a tripartite $3$-graph one of whose parts is a singleton. For every $\varepsilon>0$, there exists $\delta>0$ such that, whenever $H$ is a $3$-partite $3$-graph on parts $V_1\cup V_2\cup V_3$, each of size $n$, containing no tripartitely induced copy of $H_0$, there are subsets $V_i'\subseteq V_i$ with $|V_i'|\geq\delta n$ and $d(V_1',V_2',V_3')\in[0,\varepsilon]\cup[1-\varepsilon,1]$. Polynomial-dependence conjecture. The parameter $\delta$ may be chosen so that $\delta=\varepsilon^{O(1)}$. The proposition currently yields only the lower bound $\delta\geq 2^{-\operatorname{poly}(1/\varepsilon)}$; the conjecture asks for polynomial dependence on $\varepsilon$.