The contact-shell characterization of Reeb vector fields

Let $v$ be a non-vanishing (traversing ?) vector field on a compact $(2n+1)$-dimensional manifold $X$, and let $\xi$ be a $(2n)$-dimensional distribution normal to $v$ that admits a complex or symplectic structure. A contact-shell characterization conjecture. The vector field $v$ is a Reeb vector field of a contact form on $X$ if and only if $(X,v,\xi)$ contains no contact $(2k+1)$-shells for every $k\in[1,n]$. The preceding corollary rules out certain closed submanifolds associated with contact shells when a vector field is Reeb, motivating the necessity of the condition; the sufficiency direction is left as the conjectural part. The word “traversing” is marked as uncertain in the source, and the precise meaning of the contact shells and the induced structure on $\xi$ should be checked.