Let $x_1, \ldots ,x_n$ be positive numbers satisfying $|x_{k+1}-x_k| \leq 1$ for any $k$, and $x_1+x_2+ \ldots +x_n > \frac{n(n-1)}{2}$. For $k$ between $1$ and $n$ put
$ y_k=\frac{x_1+x_2+ \ldots +x_n}{n}-\frac{n-1}{2}+k-1 $
Then the $y_k$'s satisfy all the conditions satisfied the $x_k$'s, and $\sum_{k=1}^{n} y_k=\sum_{k=1}^{n} x_k$. Now I believe that
$ \sum_{k=1}^{n} \frac{1}{x_k} \leq \sum_{k=1}^{n} \frac{1}{y_k} $
Prove or find a counterexample.
Note that the property is true for $n=2$, because
$ \bigg(\frac{1}{y_1}+\frac{1}{y_2}\bigg)- \bigg(\frac{1}{x_1}+\frac{1}{x_2}\bigg)=\frac{(x_1+x_2)(1-(x_2-x_1)^2)}{4x_1x_2y_1y_2} $