Shapiro's Conjecture 12 for real polynomials of even degree

Let $p(x)$ be a real polynomial of even degree $n$. For a real polynomial $q$, write $\sharp_r q$ for the number of its real zeros. Shapiro's Conjecture 12.

$$\sharp_r\left[(n-1)(p'(x))^2-np(x)p”(x)\right]+\sharp_r p(x)>0.$$

The conjecture concerns the interaction between the real zeros of an even-degree polynomial and those of the differential polynomial $(n-1)(p')^2-npp”$. The paper's abstract claims a complete resolution: the assertion holds in nine mutually exclusive cases and fails in four, giving a classification in terms of root-locus properties; the precise resolution should be checked against the paper's stated cases.