A puncture of zero charge is removable

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 $Q(\rho)\to0$ as $\rho\to0^+$. Then the puncture is removable: $u$ agrees on $\dot D$ with a smooth $\tilde J$-holomorphic map $D\to\mathbb R^{2n+2}\setminus\{0\}$. (With zero charge the shifted cylinders $u(e^{-2\pi(s+it)})$ converge to constants, so $a$ stays bounded near $0$, $u$ maps a punctured neighbourhood of $0$ into a compact subset of $\mathbb R\times S$ with finite Euclidean energy, and the removable singularity theorem applies.)