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?

Let $T=S^1\times S^1 \times S^1$ be the 3-torus. Let us denote by $\alpha$ the isotopy class of the loop $S^1\times pt\times pt$ and let $\mathcal F_\alpha$ be the set of all smooth oriented foliations $F$ of $T$ 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 $T$ as total space).

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