The sum of squared logarithms inequality

Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be positive numbers. For $k\in\{1,2,\ldots,n\}$, define the standard elementary symmetric polynomials

$$e_k(y_1,\ldots,y_n)=\sum_{1\le j_1<j_2<\ldots<j_k\le n}y_{j_1}y_{j_2}\cdots y_{j_k}.$$

Sum of squared logarithms inequality. If

$$e_k(a_1,\ldots,a_n)\leq e_k(b_1,\ldots,b_n)\quad(k=1,2,\ldots,n-1),\qquad e_n(a_1,\ldots,a_n)=e_n(b_1,\ldots,b_n),$$

then

$$\sum_{i=1}^n\log^2a_i\leq\sum_{i=1}^n\log^2b_i.$$

The inequality was known for $n=2,3$ when this conjecture was formulated, and the source reports support from random sampling for small $n$; its validity for arbitrary $n\in\mathbb N$ is the open issue.