Multivariate polynomial with all coefficients positive

Let $n\geq 3$ be an integer. Consider the following polynomials :

$$f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq k}x_j\big)^2}{2}\bigg)$$

and

$$g(y_1,y_2,y_3, \ldots ,y_n)=f(y_1,y_1+y_2,y_1+y_2+y_3, \ldots, y_1+y_2+y_3+\ldots +y_n)$$

Can anybody show that all the (non-zero) coefficients of $g$ are positive ? I have checked with a computer that it is true for $n=3,4,5$.

A positive answer to the present question would immediately entail a solution to this older one.

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