Double-tower upper-bound conjecture for regularity of bounded-VC$_2$ hypergraphs

Let $\mathcal{H}$ be a $3$-graph with bounded VC$_2$ dimension, and let $\mathcal{Q}_V$ denote the vertex partition in the upper-bound theorem. Double-tower upper-bound conjecture. The bound in the upper-bound theorem can be improved to

$$\lvert\mathcal{Q}_V\rvert \leq \twr(\psi(\eta)^{-C}).$$

This conjecture asserts that the true regularity behavior for $3$-graphs with bounded VC$_2$ dimension is closer to a tower than to a double tower. The current results leave this gap open.