  Let $\mathcal{F}$ be a Morse-Bott foliation on the solid torus $T=S^1\times
D^2$ into $2$-tori parallel to the boundary and one singular central circle.
Gluing two copies of $T$ by some diffeomorphism between their boundaries, one
gets a lens space $L_{p,q}$ with a Morse-Bott foliation $\mathcal{F}_{p,q}$
obtained from $\mathcal{F}$ on each copy of $T$ and thus consisting of two
singluar circles and parallel $2$-tori. In the previous paper [O. Khokliuk, S.
Maksymenko, Journ. Homot. Rel. Struct., 2024, 18, 313-356] there were computed
weak homotopy types of the groups $\mathcal{D}^{lp}(\mathcal{F}_{p,q})$ of leaf
preserving (i.e. leaving invariant each leaf) diffeomorphisms of such
foliations. In the present paper it is shown that the inclusion of these groups
into the corresponding group $\mathcal{D}_{+}^{fol}(\mathcal{F}_{p,q})$ of
foliated (i.e. sending leaves to leaves) diffeomorphisms which do not
interchange singular circles are homotopy equivalences.
