Maximum cut and maximum bisection in random regular graphs

We prove that, for every fixed degree $d$ and a uniformly random simple $d$-regular graph $G_{n,d}$ on an even number $n>d$ of vertices, $$\mathbb{E}\operatorname{MaxCut}(G_{n,d})-\mathbb{E}\operatorname{MaxBis}(G_{n,d})=o(n).$$ In other words, requiring the two sides of a cut to have exactly the sa