Partite saturation limit conjecture

Fix $k\geq 2$, let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part, and let $F$ be a subgraph of $K_k^n$. Partite saturation limit conjecture. The limit

$$\lim_{n\to\infty}\frac{sat(K_k^n,F)}{kn}$$

exists for every fixed $k\geq 2$ and every such graph $F$. This is the partite analogue of the asymptotic-limit problem for ordinary saturation numbers; the supplied text presents it as a conjecture without resolution evidence.