A conjectural inequality of the constant terms of functions

Question

Could someone help me with the following question? This is equivalent to my previous question A conjecture about the barycenter of a polytope

Let $D$ be a differential operator defined as follows, $$\begin{equation} \begin{split} D:=\prod_{1\leq i\leq j\leq n}(\partial_{x_i}+\partial_{x_{i+1}}+\cdots\partial_{x_j})^2. \end{split} \end{equation}$$ Let $f_j(x_1,\cdots,x_n)$, $j=1,\cdots,n$, be linear functions defined as follows. $$\begin{equation} \begin{split} &f_1(x_1,\cdots,x_n)==2x_1-x_2,\\ &f_2(x_1,\cdots,x_n)=2x_2-x_3-x_1,\\ &\cdots\\ &f_j(x_1,\cdots,x_n)=2x_j-x_{j+1}-x_{j-1},\\ &\cdots\\ &f_{n-1}(x_1,\cdots,x_n)=2x_{n-1}-x_n-x_{n-2},\\ &f_n(x_1,\cdots,x_n)=2x_n-x_{n-1}.\\ \end{split} \end{equation}$$

Consider the following functions, $$\begin{equation} \begin{split} &J(y_1,\cdots,y_n,x_1,\cdots,x_n):=\prod_{i=1}^n(\sum_{k=1}^{\infty}\frac{y_i^k}{k!}f_i^{k-1}),\\ &J_1(y_1,\cdots,y_n,x_1,\cdots,x_n):=(\sum_{k=2}^{\infty}\frac{y_1^k}{k!}f_1^{k-2})\cdot\prod_{i=2}^n(\sum_{k=1}^{\infty}\frac{y_i^k}{k!}f_i^{k-1}),\\ &\cdots\\ &J_j(y_1,\cdots,y_n,x_1,\cdots,x_n):=(\sum_{k=2}^{\infty}\frac{y_j^k}{k!}f_j^{k-2})\cdot\prod_{\substack{i=1\\i\neq j}}^n(\sum_{k=1}^{\infty}\frac{y_i^k}{k!}f_i^{k-1}),\\ &\cdots\\ &J_n(y_1,\cdots,y_n,x_1,\cdots,x_n):=(\sum_{k=2}^{\infty}\frac{y_n^k}{k!}f_n^{k-2})\cdot\prod_{i=1}^{n-1}(\sum_{k=1}^{\infty}\frac{y_i^k}{k!}f_i^{k-1}).\\ \end{split} \end{equation}$$

Take $(\overline y_1,\cdots,\overline y_i,\cdots,\overline y_n)=(n+1,\cdots,i(n+1-i)+1,\cdots,n+1)$.

I am curious if the following inequality is true. $$\begin{equation} D(J-J_j)|_{(y_1,\cdots,y_n,x_1,\cdots,x_n)=(\overline y_1,\cdots,\overline y_i,\cdots,\overline y_n,0,0,\cdots,0)}>0,\,\, j=1,2,\cdots,n. \end{equation}$$

Thanks in advance!