  A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of
profinite groups if, for every group $G$ in $\mathcal{C}$ such that $w$ takes
less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite.
Detomi, Morigi and Shumyatsky established that multilinear commutator words --
and the particular words $x^2$ and $[x^2,y]$ -- have the property that the
corresponding verbal subgroup is finite in a profinite group $G$ whenever the
word takes at most countably many values in $G$. They conjectured that, in
fact, this should be true for every word. In particular, their conjecture
included as open cases power words and Engel words.
  In the present paper, we take a new approach via parametrised words that
leads to stronger results. First we prove that multilinear commutator words are
strongly concise in the class of all profinite groups. Then we establish that
every group word is strongly concise in the class of nilpotent profinite
groups. From this we deduce, for instance, that, if $w$ is one of the group
words $x^2$, $x^3$, $x^6$, $[x^3,y]$ or $[x,y,y]$, then $w$ is strongly concise
in the class of all profinite groups. Indeed, the same conclusion can be
reached for all words of the infinite families $[x^m,z_1,\ldots,z_r]$ and
$[x,y,y,z_1,\ldots,z_r]$, where $m \in \{2,3\}$ and $r \ge 1$.
