Open-subgroup triviality conjecture for word values

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

Open-subgroup triviality conjecture. If

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

then there is an open subgroup $H$ of $G$ such that

$$w(H)=1.$$

This is presented as another weaker version of the strong conciseness conjecture. Its general validity is not proved in the paper and remains open.