Mubayi–Yepremyan conjecture for random 3-graphs and linear 4-cycles

Let $G^3_{n,p}$ be the random $3$-uniform hypergraph on $n$ vertices in which each $3$-edge is present independently with probability $p$, and let $C^3_4$ be the $3$-uniform linear cycle of length $4$. Write $\mathrm{ex}(G^3_{n,p},C^3_4)$ for the largest number of edges in a $C^3_4$-free subgraph of $G^3_{n,p}$. Mubayi–Yepremyan conjecture.

$$\mathrm{ex}(G^3_{n,p},C^3_4)= \left\{ \begin{aligned} &(1+o(1))e(G^3_{n,p}),~~~~~&\text{if}~n^{-3}\ll p\ll n^{-5/3};\\ &\Theta(n^{4/3+o(1)}),~~~~~&\text{if}~n^{-5/3}\ll p\ll n^{-2/3};\\ &\Theta(pn^2),~~~~~&\text{otherwise}. \end{aligned} \right.$$

This conjecture proposes that the known lower bound is tight for $C^3_4$ across the relevant probability ranges, completing the expected asymptotic description of the random hypergraph Turán number.