Conical-vertex conjecture for minimum-degree twin-free saturated graphs

Let $\operatorname{tsat}(n,K_r,t)$ denote the minimum number of edges in a twin-free $K_r$-saturated graph on $n$ vertices with minimum degree at least $t$. A vertex is conical if it is adjacent to every other vertex. Minimum-degree conical-vertex conjecture. For all integers $r\geq 4$ and $t\geq r-2$, every extremal graph for $\operatorname{tsat}(n,K_r,t)$ has a conical vertex for large enough $n$. This is the minimum-degree analogue of the preceding conjecture and concerns the structure of extremal graphs.