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Here we used throughout, that $\delta$ can be chosen so small, that each component of the $\delta$-neighborhood (of the asymptotic orbits) contains a unique simple Reeb orbit (up to time shift), which itself is homotopically nontrivial inside the $\delta$-neighborhood. In other words the asymptotes (up to time shifts) are determined by these $C^0$-data.

The components of $v_\pm$ contributing nonzero $d\alpha$-energy are therefore parametrized by surfaces of fixed topological type (the same as for $u_\pm$), and we can (as you mentioned) choose $\delta$ small enough, to ensure that also these components coincide (as parametrized curves up to translation) with the corresponding components of $u_\pm$, as the $[u_\pm]$ are isolated in their moduli space.

Here we used throughout, that $\delta$ can be chosen so small, such that the $\delta$-neighborhood $N_\delta$ of the asymptotic orbits deformation retracts to this collection of the asymptotic orbits. In other words the asymptotes (up to time shifts) are determined by these $C^0$-data.

The components of $v_\pm$ contributing nonzero $d\alpha$-energy are therefore parametrized by surfaces of fixed topological type (the same as for $u_\pm$), and we can (as you mentioned) choose $\delta$ small enough, to ensure that also these components coincide (as parametrized curves up to translation) with the corresponding components of $u_\pm$, as the $[u_\pm]$ are isolated in their moduli space.

Edit: More on why the ends of $u_+$ coincide with the ends of $v_+$. First I should be more precise above (in paragraph 4) to say, that at that point one only knows, that the ends of $u_+$ coincide (with multiplicity) with the the ends of the simple curve underlying $v_+$.

This more precise statement should hold for the following reasons: First, all negative ends of $u_+$ can be distinguished on a $\delta$-level in the target (as $u_+$ is simple and by "unique continuation as almost complex submanifold" or perhaps more directly studying the asymptotes in coordinates). Then there is for each end $e_u$ of $u_+$ negatively asymptotic to some Reeb orbit (almost parametrized by some circle $c$ in $e_u$), a circle $c'$ in the curve $v_+$ (parametrized via $\psi_+$ by $c \subset e_u$), which lies in the same component of the $\delta$-neighborhood $N_\delta$ and is homotopic in $N_\delta$ to $c$. Then the "parametrized component" of $c'$ in $N_\delta \cap v_+$ does not leave $N_\delta$ again by the assumption on $C_0$. Thus $c'$ determines an end $e_v$ of the simple curve underlying $v_+$, which wraps around the Reeb orbit like $c'$ and thus with the same multiplicity as $e_u$. This procedure works for all ends of $u_+$ and all ends of the simple curve underlying $v_+$ are among these.

Then one proceeds with the discussion as above, using only what we know about the asymptotics of the simple curves underlying $v_\pm$.

It remains to identify the top and the bottom component as $u_\pm$ (we know already that the images have the same asymptotes). Since the asymptotes are given, and the $d\alpha$ energy is additive over the levels and converges, it follows that the sum of the $d\alpha$-energies of the simple curves underlying $v_\pm$ coincides with the $d\alpha$-energy of the $u_n$'s, which is also equal to the sum of $d\alpha$ energies of $u_\pm$. (Recall that the $d\alpha$-energy can be computed from the asymptotes).

It remains to identify the top and the bottom component as $u_\pm$ (we know already that the images have the same asymptotes). Since the asymptotes are given and the $d\alpha$-energy is computed from the asymptotes, it follows that the sum of the $d\alpha$-energies of the simple curves underlying $v_\pm$ coincides with the sum of $d\alpha$ energies of $u_\pm$. Since moreover the $d\alpha$ energy converges (and the $d\alpha$ energy is additive over the levels), the $d\alpha$ energy of $v$ is equal to the $d\alpha$-energy of the $u_n$'s, which is in turn the same as the sum of the $d\alpha$-energies of $u_\pm$.

It remains to identify the top and the bottom component as $u_\pm$ (we know already that the images have the same asymptotes). Since the asymptotes are given, and the $d\alpha$ energy is additive over the levels and converges, it follows that the sum of the $d\alpha$-energies of $v_\pm$ coincides with the $d\alpha$-energy of the $u_n$'s, which is also equal to the sum of $d\alpha$ energies of $u_\pm$. (Recall that the $d\alpha$-energy can be computed from the asymptotes).

Therefore none of the components of $v_\pm$, which contribute $d\alpha$-energy, is multiply covered. Thus at most trivial cylinders in the image of $v_\pm$ are (branched) covered. [On the other hand, condition 1d ensures now exactly that any (possibly branched) cover over the trivial cylinders with the given asymptotes $(a_i,b_i ...)$ are necessarily unbranched, and thus these components are determined as maps by the asymptotes; hence the components in $v_\pm$ which are (apriori branched) covers of trivial cylinders coincide with the unbranched covers occuring in $u_\pm$.]

It remains to identify the top and the bottom component as $u_\pm$ (we know already that the images have the same asymptotes). Since the asymptotes are given, and the $d\alpha$ energy is additive over the levels and converges, it follows that the sum of the $d\alpha$-energies of the simple curves underlying $v_\pm$ coincides with the $d\alpha$-energy of the $u_n$'s, which is also equal to the sum of $d\alpha$ energies of $u_\pm$. (Recall that the $d\alpha$-energy can be computed from the asymptotes).

Therefore all of the components of $v_\pm$, which contribute $d\alpha$-energy, are simple, i.e. they are not nontrivial branched covers of other curves. Thus at most trivial cylinders in the image of $v_\pm$ are (branched) covered. On the other hand, condition 1d ensures now exactly that any (possibly branched) cover over the trivial cylinders with the given asymptotes $(a_i,b_i ...)$ are necessarily unbranched, and thus these components are determined as maps by the asymptotes; hence the components in $v_\pm$ which are (apriori branched) covers of trivial cylinders coincide with the unbranched covers occuring in $u_\pm$.