The forbidden Berge hypergraph conjecture for the 4-cycle

Let $\mathrm{Bh}(m,F)$ denote the maximum number of edges in an $m$-vertex Berge hypergraph containing no Berge copy of the hypergraph represented by the incidence matrix $F$. Let $\1_1$ be the one-row, one-column incidence matrix and let $C_4$ denote the incidence matrix of a 4-cycle; write $\1_1\times C_4$ for their product.

Forbidden Berge hypergraph conjecture for the 4-cycle.

$$\mathrm{Bh}(m,\1_1\times C_4)=\Theta(m^2).$$

This conjecture is used in the paper to establish a classification result for the case $k=5$. Its status is not specified in the supplied text.