Sharp maximum conjecture for one-phase cones

Let $\Lambda_n:= \sup_{U \in \mathcal{S}\mathcal{C}(\mathbb{R}^n)} \lambda_1$ be the supremum of the first stability eigenvalue over singular minimizing cones in $\mathbb{R}^n$, and let $U_{n,k}$ denote the $O(n-k) \times O(k)$-invariant one-phase cone. Sharp maximum conjecture. The sharp maximum $\Lambda_n$ is attained by the cone $U_{n,n-2}$. The conjecture concerns the sharp stability bound governing eigenvalues and Jacobi-field decay in the generic regularity theory for minimizers of the Alt–Caffarelli energy. The authors establish monotonicity within the invariant family, but the asserted global sharpness over all singular minimizing cones remains open.