Polynomial dependence in the weak regularity-to-homogeneity proposition

For every $D\geq 1$ and $\varepsilon>0$, there exists $\delta>0$ such that every $3$-partite $3$-graph $H$ of slicewise VC-dimension at most $D$ that is weakly $\delta$-regular satisfies $d(H)\in[0,\varepsilon]\cup[1-\varepsilon,1]$. Polynomial-dependence conjecture. The parameter $\delta$ may be chosen so that $\delta=\varepsilon^{O(1)}$. The preceding result gives only single-exponential dependence on $\varepsilon$; the conjecture asks for a polynomial bound under bounded slicewise VC-dimension.