The eventual wheel saturation recurrence

Let $W_k$ be the wheel graph with rim cycle $C_k$, and let $(s)sat(n,H)$ denote the corresponding saturation number of a graph $H$ on $n$ vertices.

Wheel saturation recurrence. For every integer $k\geq 4$, there exists a constant $N_k$ such that, for all $n>N_k$,

$$(s)sat(n,W_k)=n-1+(s)sat(n-1,C_k).$$

The paper proves the stated results for the range treated in its theorems and conjectures that the relevant structural conclusion, and hence this recurrence, holds for all $k\geq 4$; the eventual threshold $N_k$ is asserted to exist but is not established here.