The cuboid gap-coefficient conjecture for consecutive Dirichlet eigenvalue gaps

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with piecewise smooth boundary $\partial\Omega$, and let $\lambda_i$ be the $i$-th Dirichlet eigenvalue. Let $\mathcal{S}_1(\Omega)$ denote the gap coefficient defined from the first two Dirichlet eigenvalues of the maximal-volume cuboid contained in $\Omega$. Cuboid gap-coefficient conjecture.

$$\lambda_{k+1}-\lambda_k\leq\mathcal{S}_1(\Omega)k^{\frac{1}{n}}.$$

The conjecture seeks a geometric coefficient sharper than a general universal constant. It is introduced after examples and numerical motivation; no general resolution is given.