Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$

Given a surface $\Sigma$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $\Sigma$, there exists a branched immersed disk which has constant mean curvature $H$ and boundary meeting $\Sigma$ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on $\Sigma$. Specifically, when $\Sigma$ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H \in (0, H_0)$. Instead of the heat flow used by Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou (arXiv:2012.13379), a key ingredient for extending existence across the measure zero set of $H$'s is a Morse index upper bound.