Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.

Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by at most $d$ elements.

**Can it be that $N_d = \{1\}$?**

The equivalent question in finite groups asks whether every finite group is an image of a finite subdirect product of groups generated by at most $d$ elements:

Can it be that for every finite group $G$ there exists some $r \in \mathbb{N}$ and a finite group $H$ with normal subgroups $H_1, \dots, H_r \lhd H$ such that:

$ \bigcap_{i=1}^{r} H_i = \{1\}$

For each $1 \leq i \leq r$, $H/H_i$ can be generated by at most $d$ elements.

$G$ Is a homomorphic image of $H$.