The triangular 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 $\mathcal{S}_2(\Omega)$ denote the gap coefficient defined from the first two Dirichlet eigenvalues of a maximal-volume equilateral triangle contained in $\Omega$ (in the planar case). Triangular gap-coefficient conjecture.

$$\lambda_{k+1}-\lambda_k\leq\mathcal{S}_2(\Omega)\sqrt{k}.$$

The claim is motivated by the sharp fundamental-gap result for triangles and numerical checks. The source does not provide a proof for arbitrary domains or all $k$.