Let $\mathbb{R}$ be the real field. For any homogeneous polynomial $f(X_1,\cdots,X_n)$ in $\mathbb{R}[X_1,\cdots,X_n]$, we use $S_f(X_1,\cdots,X_n)$ to denote the following homogeneous symmetric polynomial: $$S_f(X_1,\cdots,X_n)=\sum_{\sigma=[i_1,\cdots,i_n]\in S_n}f(X_{i_1},\cdots,X_{i_n}).$$ Here the sum is computed over all permutations $\sigma=[i_1,\cdots,i_n](\sigma(1)=i_1,\cdots,\sigma(n)=i_n)$ of the set $\{1,\cdots,n\}$ and the set of all such permutations is denoted $S_n.$ We say $f(X_1,\cdots,X_n)$ is good if $$S_f(a_1,\cdots,a_n)\geq 0$$ for every $(a_1,\cdots,a_n)\in \mathbb{R}^n.$ For example, when $n=3$, we have $$f(X_1,X_2,X_3)=X_3^2-X_1X_2$$ is good because$$S_f(X_1,X_2,X_3)=(X_1-X_2)^2+(X_2-X_3)^2+(X_3-X_1)^2.$$ For any $n\geq 1$, define the homogeneous polynomial of degree $n^2$ in $\mathbb{R}[X_1,X_2,\cdots,X_{2n}]$ as follow:$$\varphi_n(X_1,X_2,\cdots,X_{2n})=\prod_{\substack{1\leq i\leq n\\n+1\leq j\leq 2n}}(X_i-X_j).$$ I conjecture that $\varphi_n$ is good for any $n\geq 1$, and for this conjecture I have got the following simple results:

$(1)$It is easy to proof that $S_{\varphi_n}(X_1,X_2,\cdots,X_{2n})=0$ when $n$ is odd;

$(2)$When $n$ is even,$$S_{\varphi_2}(X_1,X_2,X_3,X_4)=4[((X_1-X_2)(X_3-X_4))^2+((X_1-X_3)(X_2-X_4))^2+((X_1-X_4)(X_2-X_3))^2].$$ So the conjecture is right for $n=2$.

I can not proof it any more for $n\geq 4$, but I believe that the conjecture is right. Would you please give me some help?