Linear asymptotic formula for $K^s_t$-saturation numbers

Let $K^s_t$ be the complete $s$-uniform hypergraph on $t$ vertices, and let $\mathrm{sat}(n,K^s_t)$ denote its saturation number on $n$ vertices. Saturation-number conjecture. For integers $s,t$ satisfying

$$2\leq s\leq t-2,$$

there exists a function $f(s,t)$ independent of $n$ such that

$$\mathrm{sat}(n,K^s_t)=\frac{t+s-2}{2}n+f(s,t).$$

This conjecture predicts an exact linear formula in $n$ with a parameter-dependent constant term for the stated range of uniformity and clique size; determining such a formula would clarify the general behavior of hypergraph saturation numbers.