One-parameter family of fast planes transverse to translations

Let $S = H^{-1}(1) \subset V_1 = \mathbb{R}^4$ be a strictly star-shaped level with a compatible complex structure $J$ on $\xi$, and let $P$ be a periodic Reeb orbit on $S$ whose Conley–Zehnder index is at least $3$. Let $u$ be an embedded fast finite-energy plane asymptotic to $P$ at $\infty$. Then for every $l \ge 1$ there are $r > 0$ and a $C^l$ map $g : \mathbb{C} \times (-r, r) \to V_1$ such that:

1. $g(\cdot, 0) = u$;
2. every $g(\cdot, t)$, $|t| < r$, is a fast finite-energy plane asymptotic to $P$ at $\infty$;
3. the map $(z, c, t) \mapsto c \cdot g(z, t) = e^{2c} g(z, t)$ is injective on $\mathbb{C} \times (-r, r) \times (-r, r)$;
4. this map has injective derivative everywhere: if $Dg(z, t)(\Delta, \delta t) + \lambda\, g(z, t) = 0$ with $\Delta \in \mathbb{C}$ and $\delta t, \lambda \in \mathbb{R}$, then $\Delta = 0$, $\delta t = 0$ and $\lambda = 0$.

Thus the planes $g(\cdot, t)$ form a one-parameter family through $u$ that is transverse to the $\mathbb{R}$-translations in the symplectization, and the small translates of its members are pairwise disjoint. This is the geometric content of Hryniewicz, Fast finite-energy planes in symplectizations and applications, Trans. AMS 364 (2012), Lemmas 6.10 and 6.20, written in a form where the $\mathbb{R}$-translation parameter is explicit.

Formalization Note: $V_1 \setminus \{0\}$ is the model of the symplectization $\mathbb{R} \times S$, in which the translation by $c$ is $y \mapsto e^{2c} y$. Item 4 is the injectivity of the derivative of the map in item 3 at $(z, c, t)$, after dividing by $e^{2c}$ and writing $\lambda = 2\,\delta c$. Since the map $g$ is only asserted to be $C^l$ on the open set $\mathbb{C} \times (-r, r)$, no uniformity as $|z| \to \infty$ is part of the statement. Non-vacuity: for the round sphere with $J = i$, $g(z, t) = (z, e^{it})$ satisfies the conclusion for $u(z) = (z, 1)$.