On the multiple conjugacy problem in group $F/{N_1\cap N_2}$

Let $F$ be a free group generated by a finite alphabet $A$. Let $N_1$ ($N_2$) be the normal closure of a finite non-empty symmetrized set $R_1$ (respectively, $R_2$) of elements in $F$. Earlier, one obtained the conditions sufficient for the solvability of the conjugacy problem in the group $F/N_1\cap N_2$. The present paper is a continuation of this research and is devoted to the solvability of the multiple conjugacy problem in $F/{N_1\cap N_2}$. In particular, we get that if $R_1\cup R_2$ satisfies the small cancellation condition $C'(1/6)$, then the multiple conjugacy problem is solvable in $F/{N_1\cap N_2}$.