Precise Gevrey regularity of invariant manifolds at parabolic infinity in restricted three-body problems

In the Sitnikov problem, let $e\in(0,1)$ denote the eccentricity, and in the restricted planar three-body problem (RPTBP), let $\mu\in(0,1/2]$ denote the mass parameter. An invariant manifold at parabolic infinity is an invariant manifold associated with the parabolic fixed point at infinity. Precise Gevrey-regularity conjecture. The parabolic infinity in the Sitnikov problem, with $e\in(0,1)$, and in the RPTBP, with $\mu\in(0,1/2]$, possesses invariant manifolds which are precisely $1/3$-Gevrey, that is, they are not $\gamma'$-Gevrey for any $0\leq\gamma'<1/3$. The preceding results establish $1/3$-Gevrey upper bounds, while numerical computations and an explicit time-periodic example motivate the conjecture that these bounds are optimal; the required lower bounds are not proved here.