Verstraete's positive discrepancy conjecture for moderately dense graphs

Let $G$ be an $n$-vertex graph of average degree $d$, where $1\leq d\leq (1/2-\varepsilon)n$.

Verstraete's conjecture.

$$\operatorname{disc}^{+}(G)=\Omega(d^{1/2}n).$$

This conjecture concerns the positive discrepancy of graphs whose average degree is at most approximately half the number of vertices. It extends the known sparse-graph lower bound and is motivated by the fact that the bound is tight for random regular graphs, while dense bipartite examples show that comparable unrestricted statements cannot hold.