Let $x\_1, \ldots, x\_n, y\_1, \ldots, y\_m$ be positive real numbers. Prove or disprove the following inequality.
\begin{equation}
\sum\_{i=1}^n\sum\_{j=1}^n \frac{1}{x\_i+x\_j} + \sum\_{i=1}^m\sum\_{j=1}^m \frac{1}{y\_i+y\_j} \geq 2\sum\_{i=1}^n \sum\_{j=1}^m \frac{1}{x\_i + y\_j}
\end{equation}

*Remark.* Computer experiments seem to suggest the positive. If this is true, (seems to me) should have been discovered in the past. By (or named after) whom and where to look for references?