Let $f:X \to Y= \mathbb P^n$ be a flat morphism. We define condition $S_k$ for such morphisms whose total space over every $k$-dimensional linear space is smooth, namely: $$f\in S_k \text{ if for every linear space $\mathbb P^k \subset \mathbb P^n$, $f^{-1}(\mathbb P^k)$ is smooth.}$$
I want to know if the following is true:
$S_i \subsetneqq S_j$ if $i<j$.
The inclusion $S_0\subset S_n$ is the well-known fact, so this can be seen as an enhancement of it. The strict inclusion is related to one of my previous question.
Thanks in advance.