Universal eigenvalue-ratio conjecture for Euclidean domains

Let $\Omega$ be a bounded domain with piecewise smooth boundary in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let $\Lambda_j$ be the $j$-th Dirichlet eigenvalue of the Laplace operator. Universal eigenvalue-ratio conjecture. For every positive integer $j$,

$$\frac{\Lambda_{j+1}+\Lambda_{j+2}+\cdots+\Lambda_{j+n}}{\Lambda_j} \leq \frac{\Lambda_2+\Lambda_3+\cdots+\Lambda_{n+1}}{\Lambda_2}.$$

The conjecture is motivated by the asymptotic fact that the left-hand ratio tends to $n$ as $j$ tends to infinity. The supplied source gives no evidence of a resolution.