As discussed in the following math overflow question, Max cut value in a random graph, the max-cut of a random graph $G(n,1/2)$ is $\frac{n^2}{8} + \Theta(n^{3/2})$ with high probability.

My question is this: Does there exist a family of graphs with relative density about $\frac{1}{2}$ with max-cut size being $\frac{n^2}{8}+ o(n^{3/2})$? If not, can we show that every graph with $\frac{1}{2}$ relative density has a cut of size $\frac{n^2}{8} + \Omega(n^{3/2})$ ?

(It is possible that any of the well-known explicit pseudorandom graphs with $\frac{1}{2}$ relative density,such as Paley graphs and etc, will satisfy this property but I haven't been able to verify whether this is the case or not by looking at a few survey papers.)