Shapiro–Shapiro conjecture for real pole placement

Let $m,p,n$ be positive integers with $n=mp$. Let $(\bar{A},\bar{B},\bar{C})$ be a minimal realization of the system represented through a coprime factorization $D^{-1}(s)N(s)$, where the matrix $\left[D(s)\mid N(s)\right]$ has the following form. The first row is

$$s^{m+p-1},s^{m+p-2},\ldots,s^2,s,1$$

and, for $1\leq j<p$, the $(j+1)$st row consists of the derivative of the $j$th row divided by $j$. Let $d(m,p)$ be the corresponding enumerative degree, and let $\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}$ denote the pole-placement map. Shapiro–Shapiro conjecture. The system is nondegenerate, and for any polynomial $\bar{\phi}$ of degree $mp$ with distinct real roots,

$$\tilde{\chi}_{(\bar{A},\bar{B},\bar{C})}^{-1}(\bar{\phi})$$

consists of exactly $d(m,p)$ real solutions. This conjecture connects a special real pole-placement problem with questions in real algebraic geometry; its resolution is not stated in the supplied text.