Classification conjecture for differential operators that are complex zero strip decreasing

Let $\phi(z) \in \operatorname{\mathcal{LP}}$ have a Weierstrass canonical product

$$\phi(z)=c e^{\alpha z-\beta z^2}\prod_n\left(1-\frac{z}{\alpha_n}\right)\exp\left(\frac{z}{\alpha_n}\right).$$

Classification conjecture. The operator $\phi(D)$ is a complex zero strip decreasing operator if and only if at least one of the following conditions holds: (i) $\beta>0$; (ii) $\beta=0$ and the product $\prod_n(1-z/\alpha_n)e^{z/\alpha_n}$ has order $\rho=1$ and type $\sigma>0$; or (iii) $\beta=0$ and the order of that product satisfies $1<\sigma\leq 2$.

The conjecture would give a complete classification of functions in the Laguerre–Pólya class whose differential operators are complex zero strip decreasing operators. The source presents this as an additional conjecture based on the preceding examples and theorem.