Ashbaugh's eigenvalue ratio conjecture for Euclidean domains

Let $\Omega$ be a bounded domain in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let $\Lambda_i$ denote the $i$-th Dirichlet eigenvalue of the Laplace operator, and let $\mathbb{B}^n$ be a ball with the same volume as $\Omega$, so that $\operatorname{Vol}(\Omega)=\operatorname{Vol}(\mathbb{B}^n)$. Ashbaugh's conjecture. The inequality

$$\frac{\Lambda_2+\Lambda_3+\cdots+\Lambda_{n+1}}{\Lambda_1}\leq n\frac{\Lambda_2(\mathbb{B}^n)}{\Lambda_1(\mathbb{B}^n)}$$

should hold. This is a higher-dimensional extension of the Payne–Pólya–Weinberger inequality; the cited source presents it as a famous conjecture, but the supplied evidence indicates that it has since been solved.