  In this article, we extend the methods from arXiv:2011.06568, where the five
dimensional analogue of the three dimensional finite energy foliations
introduced by Hofer--Wysocki--Zehnder was identified, to the case where there
the underlying (IP) contact $5$-fold admits a $6$-dimensional (IP) symplectic
filling. We show that the filling induces a moduli space of pseudo-holomorphic
curves which is itself a symplectic filling of the standard $3$-sphere, and
hence symplectomorphic to the $4$-dimensional ball. We further show that
whenever the contact form on the $5$-fold is strictly convex, then this moduli
space is a strictly convex domain, so that the induced Reeb dynamics at the
boundary is in particular dynamically convex. For the circular restricted
three-body problem, this implies that whenever the spatial dynamics (near one
of the heavy masses) is strictly convex, the holomorphic shadow of
arXiv:2011.06568 is dynamically convex; this is shown to hold for
near-integrable cases close to the Kepler problem, and with mass ratio either
zero or sufficiently close to 1.

Symplectic Geometry;Dynamical Systems