10.67 (1986)

The class $LN\mathfrak{M}_p$ of locally nilpotent groups admitting a splitting automorphism of prime order $p$ (for definition see 10.59) is a variety of groups with operators (E. I. Khukhro, Math. USSR Sbornik, 58 (1987), 119–126). Is it true that
$$LN\mathfrak{M}_p = (\mathfrak{N}_{c(p)} \cap LN\mathfrak{M}_p) \vee (\mathfrak{B}_p \cap LN\mathfrak{M}_p)$$ where $\mathfrak{N}_{c(p)}$ is the variety of nilpotent groups of some $p$-bounded class $c(p)$ and $\mathfrak{B}_p$ is the variety of groups of exponent $p$?