Unilateral $o$-Groups

For a big number of varieties $\mathcal V$ of groups close to Engelian, it is proved that a variety of lattice-ordered groups generated by all linearly ordered groups in the class $\mathcal P\mathcal V=\bigcup_{k\in\mathbf Z_+}\mathcal V^k$ does not coincide with the variety $\mathcal O_l$ of all $o$-approximable lattice-ordered groups.