Popov criterion conjecture for the scalar heat equation

Let $\beta(\theta)$ be defined for $\theta\in(0,1)$ by

$$\beta(\theta)=-\frac{1}{G_{\theta}(i\omega_1(\theta))}.$$

For a given $\theta$, let $q(\theta)>0$ be the quantity specified by the slope-direction formula. Popov criterion conjecture. For every $\beta\in(0,\beta(\theta)]$, the pair $(\beta,q(\theta))$ satisfies

$$\operatorname{Re}\bigl[G_{\theta}(i\omega)\bigr]-q(\theta)\omega\operatorname{Im}\bigl[G_{\theta}(i\omega)\bigr]\geq-\frac{1}{\beta}$$

for every $\omega\in\mathbb{R}\setminus\{0\}$. This conjecture asserts the Popov criterion throughout the indicated parameter interval; the source presents it without a proof and supports it by the stated analytical framework.