A generalized Hawaii conjecture for real zeros of differential rational functions

Let $p$ be a real polynomial of degree $n$, where $n\geqslant2$. For a real parameter $\varkappa$, let

$$Q_{\varkappa}[p](z)=\dfrac{p(z)p”(z)-\varkappa[p'(z)]^2}{p^2(z)}.$$

Generalized Hawaii conjecture. If $\varkappa>\dfrac{n-1}{n}$, then

$$Z_C(p)-Z_C(p')\leqslant Z_R(Q_{\varkappa})\leqslant Z_C(p).$$

The conjecture is proposed after the preceding Newton-type conjecture is refuted. The source notes that it is proved only in the special case $\varkappa=1$, so the general parameter range remains open there.