Balogh–et al. extremal-construction conjecture for generalized Ramsey–Turán numbers

Let $p>q\geq 3$ be integers, and let $\mathcal{G}(n;s,t)$ denote the class of constructions with parameters $s$ and $t$ used for the generalized Ramsey–Turán problem. Balogh–et al. extremal-construction conjecture. One of the asymptotically maximal graphs for $\operatorname{RT}(n,\#K_q,K_p,\alpha n)$ lies in

$$\mathcal{G}(n;q,p-q-1)$$

when $p\leq 2q-1$, and lies in

$$\mathcal{G}\left(n;\left\lceil\frac{p-1}{2}\right\rceil,\left\lfloor\frac{p-1}{2}\right\rfloor\right)$$

when $p\geq 2q$. The paper provides counterexamples, stating that the conjecture does not hold in general in both parameter ranges.