The lower-density conjecture for complex zero strip decreasing operators

Let $cphi(z) cin \operatorname{\mathcal{LP}}_1$ and let $n(t)$ denote the number of roots of $cphi(z)$ in the interval $(-t,t)$. Suppose

$$\liminf_{t \rightarrow \infty} \frac{n(t)}{t} >0.$$

Lower-density conjecture. There exists a positive constant $c_{\phi}$ such that, for any $f(z) \in \operatorname{\mathcal{LP}}(r)$ with $r>0$,

$$\phi(aD)f(z) \in \operatorname{\mathcal{LP}}\left(\sqrt{r^2-c_{\phi}a^2}\right).$$

This conjecture proposes that a positive lower density of zeros makes $\phi(aD)$ a complex zero strip decreasing operator, with a quantitative narrowing of the zero strip.