Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a *cover* if $\bigcup {\cal C} = X$. A subset $D\subseteq X$ is a *choice set* for ${\cal C}$ if $|D\cap c| = 1$ for all $c\in C$.

Ramiro de la Vega showed in his nice post that

(S): every cover has a subcover that admits a choice set.

He used the Axiom of Choice to well-order the members of the cover and construct a subcover with a choice set.

**Question.** Does the statement (S) above imply the Axiom of Choice?