On $e$-doubly stochastic matrices over hyperbolic (polynomial) systems

In the setting of a hyperbolic (polynomial) system $(\mathcal{V}, p, e)$ of degree $n$, an $n$-tuple $\mathbf{A} = \big[ a_1, a_2, \dots, a_n \big]$ is said to be an $e$-doubly stochastic $\mathcal{V}$-matrix if each $a_i$ belongs to the hyperbolicity cone, has trace $1$, and $a_1 + a_2 + \dots + a_