Finite generation conjecture for verbal subgroups

Let $w=w(x_1,\ldots,x_r)$ be a group word and let $G$ be a profinite group. Write $G_w$ for the set of $w$-values in $G$, and let $w(G)$ denote the closed subgroup generated by those values.

Finite generation conjecture. If

$$|G_w|<2^{\aleph_0},$$

then $w(G)$ is generated by finitely many $w$-values.

This is presented as a weaker version of the strong conciseness conjecture. The paper does not establish it in general, although it follows in cases where strong conciseness is known together with the relevant finite-generation argument.