Critical-regime clique-count conjecture for bounded-degree graphs

Let $n=a(r+1)+b$, and let $m$ satisfy

$$\frac{nr}{2} \ge m>a \binom{r+1}{2} + \binom{b}{2}.$$

For fixed $t$, or for $k=\sum_{t \ge 2} k_t$, consider graphs of order $n$, size $m$, and maximum degree at most $r$. Critical-regime clique-count conjecture. Any graph maximizing the chosen quantity can be represented as $(a-1)K_{r+1}+H$. This concerns the remaining, or critical, regime after the known cases in which the extremal graph is a union of copies of $K_{r+1}$ and a colex graph; the source provides no resolution of this conjecture.