A periodic orbit that bounds a disk-like global surface of section links every other periodic orbit non-trivially

Let $S=H^{-1}(1)$ be a strictly star-shaped level, and let $P=(x_P,T_P)$ be a periodic orbit (not necessarily prime) that bounds a disk-like global surface of section $D=e(\overline{\mathbb D})$ with $\partial D=e(S^1)=x_P(\mathbb{R})$. Let $Q=(x_Q,T_Q)$ be any periodic orbit on $S$ with $x_Q(\mathbb{R})\neq x_P(\mathbb{R})$. Then the loops $s\mapsto x_P(T_Ps)/|x_P(T_Ps)|$ and $s\mapsto x_Q(T_Qs)/|x_Q(T_Qs)|$ in $S^3$ have a linking number $n$ in the Gauss-integral sense of IsLinkingNumber, and $n\neq0$.

Proof idea. By uniqueness of trajectories the two images are disjoint. So $Q$ misses $\partial D$, and $Q\cap D$ is a compact subset of the interior. There $X_H$ is transverse to $D$, so the intersections are isolated and finite in number. Since $D\setminus\partial D$ is connected and $X_H$ never lies in its tangent plane, $X_H$ crosses $D$ from the same side at every point, so all intersections have the same sign. The global-section property forces at least one intersection. The linking number equals the algebraic intersection number of the loop $Q$ (traversed over $[0,T_Q]$) with the Seifert surface $D$ of $P$, which is therefore $\pm\#(Q\cap D)\neq0$, counted with the multiplicity of the covers.