inequalities - Sum of difference moduli vs. sum of modulus differences

This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.

Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, define a real $f_kleft(x_1,x_2,...,x_nright)$ as the sum

$sumlimits_{Tsubseteqleftlbrace 1,2,...,nrightrbrace ;\ left|Tright|=k} left|sumlimits_{tin T}x_t - sumlimits_{tinleftlbrace 1,2,...,nrightrbrace setminus T} x_tright|$.

We mostly care about the case of $n$ even and $k=frac n 2$; in this case, $f_kleft(x_1,x_2,...,x_nright)$ is a kind of measure for the dispersion of the reals $x_1$, $x_2$, ..., $x_n$ (more precisely, of their $frac n 2$-element sums).

Now my conjecture is that if $n$ is even and $k=frac n 2$, then

$f_kleft(x_1,x_2,...,x_nright)geq f_kleft(left|x_1right|,left|x_2right|,...,left|x_nright|right)$

for any reals $x_1$, $x_2$, ..., $x_n$.

I think I have casebashed this for $n=4$ and maybe $n=6$; I don't remember anymore - it's too long ago. Sorry. I still have no idea what to do in the general case, although my attempts at big-$n$ counterexamples weren't of much success either.