Loops converging to a periodic Reeb orbit exclude a removable puncture

Let $S=H^{-1}(1)\subset\mathbb{R}^{2n+2}$ be a strictly star-shaped level, $J$ any family of linear maps, and $u:\mathbb{C}\to\mathbb{R}^{2n+2}$ any map. Let $P=(x,T)$ be a periodic Reeb orbit on $S$, $\sigma=\pm1$, $c\in\mathbb{R}$, and $\rho_k\to0^+$. Suppose the loops $t\mapsto\mathrm{proj}\,u(\rho_ke^{-2\pi it})$ converge in $C^\infty(\mathbb{R},\mathbb{R}^{2n+2})$ to $t\mapsto x(\sigma Tt+c)$. Then the puncture $0$ of $u$ on the punctured unit disk $D\setminus\{0\}$ is not removable: $u$ does not agree on $D\setminus\{0\}$ with a smooth $\tilde J$-holomorphic map $D\to\mathbb{R}^{2n+2}\setminus\{0\}$.

This supplies the non-removability hypothesis needed to study the asymptotic behaviour of a finite-energy curve at a puncture whose boundary loops are already known to approach a periodic orbit along a sequence.

Formalization Note $C^\infty$ convergence of loops is TendstoCInfty (uniform convergence on $\mathbb{R}$ of every derivative); removability is IsRemovablePuncture H J u 0 1. No finite-energy or holomorphicity hypothesis on $u$ is assumed.