Countable covering conjecture for verbal values and group properties

Let $\Sigma$ be a property of profinite groups such that every finite group is a $\Sigma$-group, the class of $\Sigma$-groups is closed under taking subgroups, quotients and extensions, and whenever a profinite group $G$ is virtually soluble and $G/Z(G)$ is a $\Sigma$-group, its commutator subgroup $G'$ is also a $\Sigma$-group. Let $w$ be a multilinear commutator word, and let $G$ be a profinite group having countably many $\Sigma$-subgroups whose union contains all $w$-values in $G$. Countable covering conjecture. Then $w(G)$ has the property $\Sigma$. The finite-covering analogue is known, while the paper explicitly states that it does not know whether the result remains true for countable coverings.