The full-degree vertex necessity conjecture for saturation of joins

Let $F$ be a non-empty graph and let $n\geq |V(F)|+1$. Write $K_1\vee F$ for the join of $F$ with a single vertex, and call a vertex full-degree if its degree is $n-1$. An extremal graph for saturation is an $n$-vertex $K_1\vee F$-saturated graph with $\operatorname{sat}(n,K_1\vee F)$ edges.

Full-degree vertex necessity conjecture. If

$$\operatorname{sat}(n,K_1\vee F)=n-1+\operatorname{sat}(n-1,F),$$

then some extremal graph of $K_1\vee F$ contains a full-degree vertex.

The conjecture proposes a necessary structural condition for equality in the standard upper bound obtained by adjoining a universal vertex to an extremal $F$-saturated graph. The paper's discussion shows that, for the particular join $K_1\vee(K_{p-1}^-\cup K_1)$, extremal graphs have no full-degree vertex, but it does not establish the conjecture in general.