On one property of the product of non-identity formations

All groups considered are finite. The product $\mathfrak{MH}$ of the formations $\mathfrak{M}$ and $\mathfrak{H}$ is the class $\{G \mid G^{\mathfrak{H}}\in\mathfrak{M}\}$. Let $\mathfrak{MH} \subseteq \mathfrak{F}$, where $\mathfrak{F}$ is a hereditary one-generated $\omega$-saturated formation and $\mathfrak{M}$, $\mathfrak{H}$ be two non-identity formations. Suppose that $\mathfrak{MH}$ is a solubly $\omega$-saturated formation. If $\mathfrak{H} \ne \mathfrak{MH}$, then $\mathfrak{M} \subseteq \mathfrak{N}_{\omega}\mathfrak{N}$.