Local $C^\infty$-closing property for standard symplectic vector spaces

Let $({\mathbb R}^{2n},\Omega_{\textrm{standard}})$ be the standard symplectic vector space, with $n\geq 2$. A symplectic manifold has the $C^\infty$-closing property if, for every regular compact co-oriented hypersurface $M$, the induced odd-symplectic manifold has the $C^\infty$-closing property: there is a compatible symplectic form on ${\mathbb R}\times M$ such that for a Baire subset of $C^\infty(M,\mathbb R)$, the closed leaves of the induced characteristic line distribution are dense on the corresponding graph hypersurface.

Local $C^\infty$-closing property. The standard symplectic vector space $({\mathbb R}^{2n},\Omega_{\textrm{standard}})$ has the $C^\infty$-closing property for every $n\geq 2$. In particular, every symplectic manifold without boundary has the local $C^\infty$-closing property.

The conjecture concerns generic density of periodic orbits on hypersurfaces in symplectic dynamics. It is open for all $n\geq 2$, and essentially nothing is known for $n\geq 3$.