Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?

Here $d(H)$ denotes the minimal cardinality of a generating set of a group $H$. It is enough to consider the case of $G$ having a trivial Frattini subgroup.

In the language of profinite groups, an equivalent reformulation would be:

Let $F$ be a free finitely generated nonabelian prosupersolvable group. Is there a bound on the number of generators of the Sylow subgroups of $F$? (the sylow subgroups must be finitely generated by a theorem of Oltikar and Ribes).