Strong conciseness of coprime commutators in profinite groups

Let $G$ be a profinite group. The coprime commutators ${\gamma }_j^{\ast }$ and ${\delta }_j^{\ast }$ are defined as follows. Every element of $G$ is both a ${\gamma }_1^{\ast }$-value and a ${\delta }_0^{\ast }$-value. For $j\ge 2$, let $X$ be the set of all elements of $G$ that are powers of ${\gamma }_{j-1}^{\ast }$-values. An element $a$ is a ${\gamma }_j^{\ast }$-value if there exist $x\in X$ and $g\in G$ such that $a=[x,g]$ and $(\mathrm{\mid }x\mathrm{\mid },\mathrm{\mid }g\mathrm{\mid })=1$. For $j\ge 1$, let $Y$ be the set of all elements of $G$ that are powers of ${\delta }_{j-1}^{\ast }$-values. The element $a$ is a ${\delta }_j^{\ast }$-value if there exist $x,y\in Y$ such that $a=[x,y]$ and $(\mathrm{\mid }x\mathrm{\mid },\mathrm{\mid }y\mathrm{\mid })=1$. In this paper we establish the following results. A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of ${\gamma }_k^{\ast }$-values in $G$ has cardinality less than $2^{{\mathrm{\aleph }}_0}$. A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$) if and only if there is $k$ such that the set of ${\delta }_k^{\ast }$-values in $G$ has cardinality less than $2^{{\mathrm{\aleph }}_0}$.