Loops near a puncture of nonzero charge converge to a Reeb orbit

Let $S=H^{-1}(1)\subset\mathbb R^{2n+2}$ be a strictly star-shaped level with contact form $\lambda=\lambda_0|_S$ and $d\lambda$-compatible complex structure $J$ on $\xi$, and let $u=(a,w):\dot D=D\setminus\{0\}\to\mathbb R\times S\cong\mathbb R^{2n+2}\setminus\{0\}$ be a finite-energy $\tilde J$-holomorphic curve on the punctured unit disc. Write $Q(\rho)=\int_0^1 w(\gamma_\rho(t))^*\lambda$ for the action of the clockwise loop $\gamma_\rho(t)=\rho e^{-2\pi i t}$, and call $|z|\,|\nabla u(z)|\le C$ for $0<|z|<r$ the gradient bound near the puncture, where $|\nabla u|=\|Da\|+\|Dw\|$ ($\mathbb R$-invariant). Assume the gradient bound near the puncture and $Q(\rho)\to Q\neq0$. Then for every sequence $\rho_k\to0^+$ there are a subsequence $\rho_{k_j}$, a periodic Reeb orbit $(x,|Q|)$ (period $|Q|$) and $c\in\mathbb R$ such that the loops $t\mapsto w(\rho_{k_j}e^{-2\pi i t})$ converge in $C^\infty(\mathbb R)$ to $t\mapsto x(Qt+c)$. (The gradient bound gives $C^\infty_{loc}$-compactness of the shifted cylinders $u(\rho_k e^{-2\pi(s+it)})-a(u(\rho_k))$; the limit has zero $d\lambda$-energy, hence is an orbit cylinder, and its loop action is $Q$.)