Tower-growth conjecture for hereditary 3-uniform hypergraph properties

Let $\mathcal{H}$ be a hereditary property of $3$-uniform hypergraphs with infinite $\operatorname{VC}_2$-dimension. Let $L_{\mathcal{H}}(\epsilon_1,\epsilon_2)$ denote the associated regularity growth function, and let $Tw$ denote the tower function. Tower-growth conjecture. There exist a function $\epsilon_2:\mathbb{N}\rightarrow (0,1]$ and a constant $C>0$ such that

$$L_{\mathcal{H}}(\epsilon_1,\epsilon_2)\geq Tw(\epsilon_1^{-C}).$$

This conjecture proposes that the lower bound in the unbounded-$\operatorname{VC}_2$ range can be improved from exponential to at least tower growth; the paper suggests that techniques from its proof combined with bipartite lower-bound constructions for graph regularity may establish it. The source does not give evidence of a resolution.