Let $(M^3,g)$ be a compact Riemannian $3$-manifold with boundary. In a paper by L. Ambrozio, he considers the set $\mathcal{F}_M$ of all immersed disks in $M$ whose boundaries are curves in $\partial M$ that are homotopically non-trivial. Two quantities are defined:
$$\mathcal{A}(M,g) = \inf_{D \in \mathcal{F}_M} \operatorname{Area}(D) \quad \text{ and } \quad \mathcal{L}(M,g) = \inf_{D \in \mathcal{F}_M} \operatorname{Length}(\partial D).$$
He then proves the following theorem:
If $\mathcal{F}_M \neq \emptyset$ and $\partial M$ is mean-convex (positive mean curvature), then $$\frac{1}{2} \mathcal{A}(M,g) \inf_M R_M + \mathcal{L}(M,g) \inf_{\partial M} H^{\partial M} \leq 2 \pi,$$ where $R_M$ is the scalar curvature of $M$ and $H^{\partial M}$ is the mean curvature of $\partial M$. Moreover, if equality holds, then the universal cover of $(M,g)$ is isometric to a cylinder $(\Sigma_0 \times \mathbb{R}, g_0 + dt^2)$, where $(\Sigma_0, g_0)$ is a disk with constant Gaussian curvature $\inf_M R_M/ 2$ and $\partial \Sigma_0$ has constant geodesic curvature $\inf_{\partial M} H^{\partial M}$ in $\Sigma_0$.
My question is whether the converse of the last statement in this theorem holds. Namely: if the universal cover of $(M, g)$ is isometric to a cylinder $(\Sigma_0 \times \mathbb{R}, g_0 + dt^2)$, where $(\Sigma_0, g_0)$ is a disk with constant Gaussian curvature $\inf_M R_M/ 2$ and $\partial \Sigma_0$ has constant geodesic curvature $\inf_{\partial M} H^{\partial M}$ in $\Sigma_0$, does equality hold?