Spherical cap conjecture for compact CMC surfaces bounded by a circle

Let $\Sigma$ be a compact surface in $\mathbb{R}^3$ with non-empty circular boundary and nonzero constant mean curvature. Spherical cap conjecture. If either (i) $\Sigma$ has genus zero and is immersed, or (ii) $\Sigma$ is embedded, then $\Sigma$ is a spherical cap. The genus-zero immersed case and the embedded case extend classical uniqueness results for closed constant-mean-curvature surfaces; the claim remains open for compact surfaces with boundary, although immersed examples of nonzero genus are known.