A product variety of groups with distributive lattice

By a variety of $A$-groups is meant a locally finite variety of groups whose nilpotent groups are abelian. It is shown that if $\mathfrak{U}$ is a variety of $A$-groups and $\mathfrak{B}$ is a locally finite variety whose lattice of subvarieties is distributive and the exponents of $\mathfrak{U}$ and $\mathfrak{B}$ are coprime, then the lattice of subvarieties of the product variety $\mathfrak{U}\mathfrak{B}$ is distributive.