Homogeneous positivity conjecture for the Stanley polynomial

Let $F$ and $\phi$ be the polynomial and map defined in the preceding setup, and define

$$\rho(\{y\},\beta)=(-1)F(\phi(\{y\}),\beta).$$

Homogeneous positivity conjecture. The polynomial $\rho(\{y\},\beta)$ has non-negative coefficients as a homogeneous polynomial in $y_i$ and $\beta$, and it has degree $7$ in $\beta$. The source gives this as a conjectural positivity observation related to the Stanley-polynomial construction; the notation and any precise domain for the $y_i$ are inherited from the preceding definitions.