Räty–Sudakov–Tomon surplus conjecture for graphs far from clique unions

Let $G$ be a regular graph on $n$ vertices, with max-cut surplus $\mathsf{sp}(G)$. Say that $G$ is $\epsilon$-far from a graph $H$ if no graph isomorphic to $H$ can be obtained from $G$ by adding or removing at most $\epsilon |V(G)|^2$ edges.

Räty–Sudakov–Tomon conjecture. For any $\epsilon>0$, there exists a $c>0$ such that, if $G$ is $\epsilon$-far from every disjoint union of cliques, then

$$\mathsf{sp}(G) \geq c n^{5/4}.$$

This conjecture predicts a polynomial improvement over Edwards' bound for regular graphs that are far from the natural low-surplus examples. The supplied text gives no resolution; the cited work establishes a related lower bound with a logarithmic loss in a restricted very dense regular setting.