Piecewise growth-rate conjecture for $\operatorname{ex}(G_{n,p}^{(3)},C_4^{(3)})$

Let $G_{n,p}^{(3)}$ be the random 3-uniform hypergraph on $n$ vertices, let $C_4^{(3)}$ be the 3-uniform expansion of the 4-cycle, and let $\operatorname{ex}(G_{n,p}^{(3)},C_4^{(3)})$ be the largest number of edges in a $C_4^{(3)}$-free subgraph. Piecewise $C_4^{(3)}$ growth-rate conjecture. One should have

$$\operatorname{ex}\left(G_{n,p}^{(3)}, C_{4}^{(3)}\right) = \begin{cases} (1+o(1))e(G_{n,p}^{(3)}), & \operatorname{if } {n^{-3}\ll p \ll n^{-5/3}},\\ \Theta(n^{4/3+o(1)}), & \operatorname{if } {n^{-5/3}\ll p \ll n^{-2/3} },\\ \Theta(pn^{2}),& \operatorname{otherwise.} \end{cases}$$

This gives a complete conjectural description of the random Turán number, including the $n^{4/3}$ middle regime. The paper presents this as an open problem; the displayed formula is the fullest version of the preceding conjectural growth-rate statement.