The maximal-imaginary-part conjecture for roots of SNN polynomials

Let $M_d(t)$ be the polynomial associated with degree $d$, and let $\gamma_d$ be a root of $M_d(t)$ with largest norm. An SNN polynomial is a degree-$d$ polynomial satisfying Stanley's non-negativity condition. Maximal-imaginary-part conjecture. The quantity $|\operatorname{Im}(\gamma_d)|$ is maximal among the imaginary parts of all roots of degree-$d$ SNN polynomials. Experimental data for SNN polynomials of degree at most seven support this conjecture; it is proved in the source when $d=2$.