A strengthened equality conjecture for real zeros of logarithmic derivatives

Let $p$ be a real polynomial with $Z_{\mathbb{C}}(p)=2m>0$. For each real $\sigma$, let $p_{\sigma}$ be the polynomial defined in the source, and write

$$2m_{\sigma}=Z_{\mathbb{C}}(p_{\sigma})\leq 2m.$$

Strengthened Hawaii conjecture. There exists $\sigma\in\mathbb{R}$ such that

$$Z_{\mathbb{R}}\left(Q[p]\right)=2m-2m_{\sigma}+Z_{\mathbb{R}}\left(Q[p_{\sigma}]\right).$$

Moreover, if there exist $x,\sigma\in\mathbb{R}$ with $Q[p_{\sigma}](x)>0$, then $\sigma$ can be chosen so that $2m_{\sigma}<2m$.

This is presented as a stronger fact following a proven lower bound obtained using Cauchy indices. Its status is left open because the supplied text gives no resolution evidence.