McKinley–Spiro random Turán conjecture for bipartite graphs

Let $F$ be a graph with

$$\mathrm{ex}(n,F)=\Theta(n^\alpha)$$

for some $\alpha\in(1,2]$. Write $G_{n,p}$ for the random graph with edge probability $p$, and let $d_2(F)$ denote the 2-density parameter appearing in the conjecture. A sequence of events holds asymptotically almost surely if its probability tends to $1$ as $n\to\infty$, and $f(n)\ll g(n)$ means $f(n)/g(n)\to0$. McKinley–Spiro's conjecture. Asymptotically almost surely,

$$\mathrm{ex}(G_{n,p},F)=\begin{cases}\max\{\Theta(p^{\alpha-1}n^\alpha),\,n^{2-1/d_2(F)}(\log n)^{\Theta(1)}\}&p\gg n^{-1/d_2(F)},\\(1+o(1))p\binom{n}{2}&n^{-2}\ll p\ll n^{-1/d_2(F)}. \end{cases}$$

The conjecture predicts three regimes for the random Turán number, including a middle regime that is essentially independent of $p$. It generalizes the behavior known in several bipartite graph cases, but the general statement remains open.