Saturation-number formula for joins of cliques and paths

Let $P_k$ be the path on $k$ vertices, let $K_s$ be the complete graph on $s$ vertices, and let $sat(n,H)$ denote the minimum number of edges in an $H$-saturated graph on $n$ vertices. Let $a_k$ be defined as in equation $(1.1)$, with $k,s\geq 3$ and $n\geq a_k+s$. Saturation-number conjecture.

$$sat(n,K_s\vee P_k)=\binom{s}{2}+s(n-s)+sat(n-s,P_k).$$

This conjecture extends the known saturation-number results for $k\in\{1,2\}$, $s=1$, and $s=2$, and is motivated by Proposition 5.4, which establishes the displayed quantity as an upper bound. The equality remains open in the stated range.