Clique-counting conjecture for bounded-degree graphs with a fixed number of edges

Let $t\geq 3$, let $G$ be a graph with $m$ edges and maximum degree at most $r$, and write

$$m=a\binom{r+1}{2}+b,\qquad 0\leq b<\binom{r+1}{2}.$$

Here $k_t(G)$ denotes the number of copies of $K_t$ in $G$, $K_{r+1}$ is the complete graph on $r+1$ vertices, and $\mathcal{C}(b)$ is the colex graph with $b$ edges. Bounded-degree edge-resource clique-counting conjecture. One has

$$k_t(G)\leq k_t\bigl(aK_{r+1}\cup\mathcal{C}(b)\bigr).$$

The conjecture refines the paper's main theorem, which proves the analogous assertion for the total number of cliques. The source attributes this refinement to the authors' earlier work, but gives no resolution status, so it remains open here.