Nested-clique extremal conjecture for the inverse Turán problem

For positive integers $r_1\geq\dots\geq r_\ell$, let $K(r_1,\dots,r_\ell)$ be the multigraph consisting of nested copies of $K_{r_i}$: on vertex set $[r_1]$, overlay a copy of $K_{r_i}$ on $[r_i]$ for every $i$. Thus, the maximum edge-weight is $\ell$ provided every $r_i\geq 2$. Nested-clique extremal conjecture. For every $k$, there exist positive integers $r_1\geq\dots\geq r_\ell$ such that $K(r_1,\dots,r_\ell)$ is extremal for $\mathcal{E}_{K_t}^*(k)$. The conjecture proposes that nested complete multigraphs always attain the extremal value in the inverse Turán problem, beyond the cases where $k=\operatorname{ex}(K_n,K_t)+1$; the precise values of $\mathcal{E}_{K_t}(k)$ and $\mathcal{E}_{K_t}^*(k)$ are otherwise not known in general.