On the torsion in the group $F/[M,N]$ in the case of combinatorial asphericity of the groups $F/M$ and $F/N$

Let $F$ be a nonabelian free group with basis $A$, and let $M$ and $N$ be the normal closures of sets $R_M$ and $R_N$ of words in the alphabet $A^{\pm 1}$. As is known, there is no torsion in the group $F/[N,N]$; however, in general, a torsion in $F/[M, N]$ is possible. In the paper by Kuz'min and Hartley (1991), it was proved that if $R_M=\{v\}$, $R_N=\{w\}$, and the words $v$ and $w$ are not proper powers in $F$, then there is no torsion in $F/[M,N]$. In this paper, we obtain a sufficient condition for the absence of torsion in $F/[M,N]$, which enables us to generalize the result of Kuz'min and Hartley to arbitrary words $v$ and $w$.