Yau's variant of the Penrose conjecture

Let $\alpha\geq 0$, and let $(M^3,g)$ be an asymptotically flat manifold with nonnegative scalar curvature containing an outermost minimal surface $\Sigma$ with induced metric $\gamma$. Let

$$\lambda_0^{\alpha}(\Sigma,\gamma)<\lambda_1^{\alpha}(\Sigma,\gamma)\leq\cdots$$

be the eigenvalues of the operator $-\Delta_{\Sigma}+\alpha K$, where $K$ is the Gauss curvature of $(\Sigma,\gamma)$. Yau's variant of the Penrose conjecture. The ADM mass should satisfy

$$m_{\operatorname{ADM}}(M,g)\geq\sqrt{\frac{1}{2\lambda_1^{\alpha}(\Sigma,\gamma)}}.$$

This proposed inequality is a variant of the Penrose inequality involving the first positive eigenvalue of a curvature-weighted Laplacian. The paper's abstract states that it gives a counterexample to this recently conjectured variant, so the conjecture is refuted.