Piecewise asymptotic conjecture for random 3-uniform linear 4-cycles

Let $G_{n,p}^{(3)}$ be the random 3-uniform hypergraph on $n$ vertices in which each triple is present independently with probability $p$, let $C_4^{(3)}$ be the 3-uniform expansion of the 4-cycle, and let $e(G_{n,p}^{(3)})$ denote the number of edges of $G_{n,p}^{(3)}$. Write $\mathrm{ex}(G_{n,p}^{(3)},C_4^{(3)})$ for the maximum number of edges in a $C_4^{(3)}$-free subgraph.

Piecewise asymptotic conjecture for random 3-uniform linear 4-cycles.

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

This conjecture gives the proposed full behaviour of the random Turán number, refining the preceding growth-rate conjecture into three density regimes. The paper presents it as an open conjecture; its first regime corresponds to retaining almost all random edges, the middle regime to the $n^{4/3}$ scale, and the final regime to the natural $pn^2$ scale.