Day's conical-vertex conjecture for extremal $K_r$-saturated graphs

Let $r\geq 4$ and $t\geq r-2$ be integers. Let $C(r,t)$ denote the family of graphs defined by $C(r,t)=C(r,t,-c(r,t))$, where $c(r,t)$ is the corresponding extremal constant for $K_r$-saturated graphs with minimum degree at least $t$. A conical vertex is a vertex adjacent to every other vertex. Day's conjecture. Every graph in $C(r,t)$ has a conical vertex.

This conjecture is equivalent to the assertion that, for fixed $r\geq 4$ and $t\geq r-2$ and sufficiently large $n$, every extremal graph for $sat(n,K_r,t)$ has a conical vertex. The source reports Day's speculation and states that the author believes it to be true, but provides no resolution.