Could someone help me with the following conjecture? Thanks a lot!

Suppose I have a polytope $\Delta$ in $\mathbb R^n (n\geq 2)$  with coordinates $(x_1,x_2,\cdots,x_n)$ defined by linear inequalities as follows.
\begin{equation}
\begin{split}
\Delta=\{(x_1,\cdots,&x_n)|0\leq x_1\leq n+1,\,0\leq x_2\leq 2(n-1)+1,\,\cdots,\\
&0\leq x_i\leq i(n+1-i)+1,\,\cdots,0\leq x_n\leq n+1,\\
&f_1\geq 0,\,f_2\geq 0,\,\cdots,f_n\geq 0\},
\end{split}
\end{equation}
where, under the convention $x_0=x_{n+1}=0$, 
\begin{equation}
\begin{split}
&f_1(x_1,\cdots,x_n)=2x_1-x_2-x_0=2x_1-x_2,\\
&f_2(x_1,\cdots,x_n)=2x_2-x_3-x_1,\\
&\cdots\\
&f_j(x_1,\cdots,x_n)=2x_j-x_{j+1}-x_{j-1},\\
&\cdots\\
&f_{n-1}(x_1,\cdots,x_n)=2x_{n-1}-x_n-x_{n-2},\\
&f_n(x_1,\cdots,x_n)=2x_n-x_{n+1}-x_{n-1}=2x_n-x_{n-1}.\\
\end{split}
\end{equation}


Let $\rho(x_1,\cdots,x_n)$ be the density function over $\Delta$ defined as follows,
$$\rho(x_1,\cdots,x_n):=\prod_{1\leq i<j\leq n+1}((x_i-x_{i-1})-(x_j-x_{j-1}))^2.$$
Here we also use the the convention that $x_0=x_{n+1}=0$.

The barycenter $\overline X=(\overline x_1,\cdots,\overline x_n)$ of $\Delta$ with respect to $\rho$ is given by the following formula
$$\overline x_i:=\frac{\int_{\Delta}x_i\cdot\rho(x_1,\cdots,x_n)\, dx_1dx_2\cdots dx_n}{\int_{\Delta}\rho(x_1,\cdots,x_n)\, dx_1dx_2\cdots dx_n}$$
where $i=1,\cdots,n$.

Then I would like to know if the following conjecture is true. (I have check the case $n=2,3,4$ by mathematica but do not have any clue how to prove it or disprove it.)
$${\bf Conjecture}:\overline x_i>i(n+1-i)\,\,{\rm for\,}i=1,\cdots,n.$$