Stanley positivity conjecture for an eight-parameter family

Let $\phi$ be the displayed map from the five auxiliary variables into the specified linear space, and write $\{m,n,r\}=(m_1,m_2,m_3,n_1,n_2,n_4,r_1,r_2)$. Stanley positivity conjecture. The expression

$$g:=(-1)[\bm G_{\star}(\phi(\{m,n,r\}),\alpha-1)]$$

is a polynomial in $\alpha$ with non-negative coefficients for every

$$(m_1,m_2,m_3,n_1,n_2,n_4,r_1,r_2)\in\mathbb Z_{\a0\geq0}^{8}$$

satisfying $r_2n_4=0$. The source explains that this remains conjectural because the proposed Stanley sum has not been proved to capture the Jack Littlewood–Richardson coefficients.