TITLE: The sum of squared logarithms conjecture
QUESTION [40 upvotes]: I am searching for the first proof of (or counterexample to) the following conjecture. 
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all 
$k\in\{1,\ldots, n-1\}$ 
it holds
$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$
and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$
it follows 
$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$
Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by
$x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.
Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at
https://www.uni-due.de/mathematik/ag_neff/log_conjecture
Immediately after Lev Borisov's sketch of a proof idea below, Lev Borisov, Suvrit Sra, Christian Thiel and myself agreed to work out the details and to write together a complete and self-contained paper on the sum of squared logarithm conjecture and relations to other topics which can be found at: http://arxiv.org/abs/1508.04039
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As announced in my first post, the prize winner is Lev Borisov.

REPLY [6 votes]: The article http://arxiv.org/abs/1410.2706 by Neff and Pompe gives a criterion, for which functions the partial derivative with respect to the elementary symmetric polynomials is nonnegative (and checks it for log^2). 
The difficult part here is not to show that the partial derivatives are nonnegative, but to show that this already implies the conjecture.
The reason is that the function, whose partial derivatives then are checked, is defined on a not-so-nice set, namely 
S={(e_1(x_1,...,x_n),e_2(x_1,...,x_n),...,e_n(x_1,...,x_n)); x_i>0}, 
which is, e.g., non-convex. (And it is not difficult to find functions with nonnegative partial derivatives and sufficiently "strange" domain, which are not monotone.) Maybe S has some structure that can be used, but I do not see it.
A large part of the article mentioned (probably for this reason) is concerned with the construction of curves, along which the nonnegativity of partial derivatives can be utilized in order to obtain monotonicity. 
Difficulties with the "only real"-approach appear at the boundary of the set S - there, invoking the implicit function theorem becomes impossible.
Of course, by his clever idea to consider the question over the complex numbers, the function in Lev Borisov's solution is defined on a much larger (and convex) set and thereby this problem is avoided entirely. Congratulations.