The max-cut local concentration conjecture for random graphs

Let $G_{n,p}$ be the binomial random graph. An ordinary max cut is a partition of the vertex set into two parts attaining the maximum possible number of crossing edges. Max-cut local concentration conjecture. If

$$p\gg n^{-1}\log n,$$

then with high probability no ordinary max cut of $G_{n,p}$ contains all, or even $51\%$, of the edges incident with any vertex. The source notes that a weaker bound is known for $r\geq 4$, while the asserted range is not established there; for $r=3$ even the bound $p>Cn^{-1/2}$ was not known.