Stability of closed characteristics on compact convex hypersurfaces in $\mathbb{R}^6$

Let Σ ⊂ ℝ6 be a compact convex hypersurface. We prove that if Σ carries only finitely many geometrically distinct closed characteristics, then at least two of them must have irrational mean indices. Moreover, if Σ carries exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.