Shapiro's even-degree weighted Hawaiian inequality

Let $p(x)$ be a real polynomial of even degree $k$. Define

$$G_1(x)=(k-1)(p'(x))^2-kp(x)p”(x).$$

Even-degree weighted Hawaiian conjecture.

$$\sharp_r(G_1(x))+\sharp_r(p(x))>0.$$

The inequality is stated as an additional claim; the source notes that the corresponding assertion is trivial for odd-degree polynomials but gives no resolution for even degree.