Strict fundamental-gap inequality for higher-dimensional convex domains

Let $\Omega\subset\mathbb{R}^n$ be a convex domain, and assume $n>1$. Write $\xi(\Omega)$ for the gap function, namely the difference between the first two Dirichlet eigenvalues of $\Omega$. Strict fundamental-gap conjecture.

$$\xi(\Omega)>3\pi^2.$$

The fundamental gap conjecture identifies $3\pi^2$ as the sharp lower bound for convex domains, approached by thin tubular domains and attained in one dimension. This conjecture asks whether the minimum is strict, and hence unique to the one-dimensional case.