Polynomial graph-partition conjecture for bounded-VC$_2$ hypergraphs

Assume for simplicity that $\mathcal{H}$ is tripartite with parts $X,Y,Z$, each of size $n$. An $\varepsilon$-graph partition consists of partitions

$$X\times Y=E_1\sqcup\dots\sqcup E_M,\quad X\times Z=F_1\sqcup\dots\sqcup F_M,\quad Y\times Z=G_1\sqcup\dots\sqcup G_M,$$

and a function $f:[M]^3\to\{0,1\}$ such that for all but $\varepsilon n^3$ triples $(x,y,z)\in X\times Y\times Z$, if $xy\in E_i$, $xz\in F_j$, and $yz\in G_k$, then

$$xyz\in E(\mathcal{H})\Longleftrightarrow f(i,j,k)=1.$$

Polynomial graph-partition conjecture. If $\mathcal{H}$ has bounded VC$_2$ dimension, then it has an $\varepsilon$-graph partition with $M=\operatorname{poly}(1/\varepsilon)$ many graphs. This would sharpen the tower-type bound currently obtained for such graph partitions; polynomial bounds remain open.