Rigidity conjecture for complete balanced Hermitian manifolds

Let $(M,\omega)$ be a complete balanced Hermitian manifold of complex dimension $n$, and suppose that the equality case in the first-eigenvalue estimate of Theorem 1.1 holds, so that $D=\frac{\pi}{\sqrt{K}}$. The theorem's Kähler rigidity conclusion identifies $(M,\omega)$, up to scaling, with $(\mathbb{CP}^1,\omega_{\mathrm{FS}})$. Rigidity conjecture. The same conclusion remains valid without assuming that $\omega$ is Kähler: $(M,\omega)$ is isometric to $(\mathbb{CP}^1,\omega_{\mathrm{FS}})$ up to a scaling. This asks whether the rigidity phenomenon for equality in the first-eigenvalue estimate persists in the more general, possibly non-Kähler, balanced Hermitian setting.