This question is somewhat related to Differential inclusions for distributions. but I am asking for something rather more specific, so I hope it is alright to leave this as a separate, new question.

Let $M$ be a smooth manifold, then given a one-form $\omega$ on $M$, Frobenius' theorem gives a simple way to test whether the distribution defined by the kernel of $\omega$ in $TM$ is integrable.

Now let $F\subset T^*M$ be a subset of the cotangent bundle such that the restriction of the canonical projection $\pi$ is onto, and such that $\pi^{-1}(p)$ is an open convex cone for every $p$ (I don't know whether this condition will affect the answer; I'm just including the information on what I know). I am interested in knowing conditions which will guarantee that there exists (locally) an integrable distribution in $F$.

Trvially some restrictions must apply. One may imagine that $F$ is in some sense not continuous, such that for any $\omega_p\in F_p$, there does *not* exist any smooth extension $\omega$ to any neighborhood of $p$. An example would be taking $M$ to be $\mathbb{R}^2$, and $F_p$ to be the first quadrant for all $p\neq 0$, and the second quadrant for $p = 0$.

So one specific question is: is this lack of freedom the only difficulty? Is the following statement true?

Suppose $F$ has the property that, for any $p$, and any $\omega_p \in F_p$, there exists some smooth one-form $\omega$ that is a section of $F$, such that $\omega |_p = \omega_p$, and $d\omega |_p = 0$, then for any $q\in M$, we can find an open neighborhood $U$ of $q$ and some one-form $\eta$ over $U$ such that $\eta$ is a section of $F|_U$ and $d\eta = 0$.

Feel free to ask for clarifications.