Let $M$ be a smooth manifold and $\mathcal{F}=\{\mathcal{F}_m\}_{m\in M}$ be a (regular) smooth foliation of $M$. The leaves $\mathcal{F}_m$ are smoothly immersed and moreover weakly embedded submanifolds. What can be said about
$$S:= \bigcup_{m\in M}\{m\}\times \mathcal{F}_m \subset M\times M$$
in general? Is $S$ also a leaf of some foliation of $M\times M$? If not, is $S$ still a weakly embedded submanifold of $M\times M$ or, if not, at least an immersed submanifold of $M\times M$?
Edit 1: $S\subset M\times M$ is an immersed submanifold if $S$ has a topology and smooth structure making the inclusion map $S\hookrightarrow M\times M$ an injective immersion. If $S$ is an immersed submanifold, $S$ is a weakly embedded submanifold if for every smooth manifold $Q$ and smooth map $F:Q\to M\times M$ satisfying $F(Q)\subset S$, $F$ is also smooth when viewed as a map $F:Q\to S$.
I do not know how standard this terminology is, but it is introduced on pp. 113-114 of Lee's "Introduction to Smooth Manifolds", 2nd edition. Lee says that some other authors call weakly embedded submanifolds "initial submanifolds". See Theorem 5.33 on p. 115 of Lee's book for one fact about these, and Theorem 19.17 on p. 500 for a proof that every leaf of a smooth foliation is weakly embedded.
Edit 2: To be more explicit, I am looking for a definitive answer to the following multiple choice question.
Which of the following mutually exclusive statements is true?
A. In general, $S$ is a weakly embedded submanifold of $M\times M$.
B. In general, $S$ is an immersed but not weakly embedded submanifold of $M\times M$.
C. In general, $S$ does not admit the structure of an immersed submanifold of $M\times M$.
