Shapiro's weighted Hawaiian conjecture for the first Pólya polynomial

Let $p(x)$ be a real polynomial of degree $k$ with simple real zeros. Define

$$G_1(x)=(k-1)(p'(x))^2-kp(x)p”(x).$$

Weighted Hawaiian conjecture.

$$\sharp_r(G_1(x))\leq\sharp_{nr}p(x).$$

The claim extends the settled Hawaiian conjecture to the first weighted expression $G_1$. The source gives no resolution.