Carlson et al.'s Shearer-type surplus conjecture for clique-free graphs

Let $r\ge 3$ be fixed. A graph is $d$-degenerate if every subgraph has a vertex of degree at most $d$, and $K_r$ denotes the complete graph on $r$ vertices. For a graph with $m$ edges, let its surplus be the quantity measuring the excess of its maximum cut over the baseline determined by its number of edges.

Carlson et al.'s conjecture. Every $K_r$-free $d$-degenerate graph with $m$ edges has surplus

$$\Omega_r\left(\frac{m}{\sqrt{d}}\right).$$

Carlson et al. proposed this as an analogue of Shearer's bound for $K_r$-free graphs and showed that it would imply the conjecture of Alon, Bollobás, Krivelevich and Sudakov on surplus in $H$-free graphs. The supplied material does not state whether this conjecture has been resolved.