The sum of squared logarithms inequality in n dimensions

For $n\in\mathbb{N}$, let $\lambda_1,\lambda_2,\ldots,\lambda_n,\mu_1,\mu_2,\ldots,\mu_n\in\mathbb{R}_+$, and let

$$e_k(X_1,X_2,\ldots,X_n)=\sum_{1\leq j_1<j_2<\cdots<j_k\leq n}X_{j_1}X_{j_2}\cdots X_{j_k},\qquad k=1,\ldots,n,$$

be the elementary symmetric polynomials. The sum of squared logarithms inequality. If

$$e_k(\lambda_1,\lambda_2,\ldots,\lambda_n)\leq e_k(\mu_1,\mu_2,\ldots,\mu_n),\qquad k=1,\ldots,n-1,$$

and

$$e_n(\lambda_1,\lambda_2,\ldots,\lambda_n)=e_n(\mu_1,\mu_2,\ldots,\mu_n),$$

then

$$\sum_{k=1}^n\log^2\lambda_k\leq\sum_{k=1}^n\log^2\mu_k.$$

The inequality extends the corresponding results established in dimensions two and three and is relevant to the analysis of logarithmic strain energies and their convexity properties. The general $n$-dimensional case is presented as a conjecture, while the two- and three-dimensional cases are known.