Let $\mathbb T^3=\mathbb T\times \mathbb T \times \mathbb T$ be the 3-torus. Let us denote by $\alpha$ the free-homotopy class of the loop $\mathbb T\times 0\times 0$ and let $\mathcal F_\alpha$ be the set of all smooth oriented foliations $F$ of $\mathbb T^3$ by circles such that every leaf of $F$ lies in the class $\alpha$ (in particular $F$ yields a smooth principal $S^1$-bundle over the 2-torus having $\mathbb T^3$ as total space).

Question: Is the space $\mathcal F_\alpha$ connected?