The converse Laguerre-operator characterization for shifted Laguerre–Pólya functions

Let $\phi(x)$ be a real entire function in the shifted Laguerre–Pólya class $\mathcal{SL\text{-}P}$. For each $d\in\mathbb{N}$, suppose there exists $N_3(d)$ such that

$$L_k(\phi^{(n)}(x))\geq 0$$

for all $0\leq k\leq d$, $n\geq N_3(d)$, and $x\in\mathbb{R}$, where the operators $L_k$ are defined by

$$|\phi(x+iy)|^2=\sum_{k=0}^{\infty}L_k(\phi(x))y^{2k}.$$

The converse Laguerre-operator conjecture. The converse direction of this characterization is true: these eventual nonnegativity conditions imply that $\phi$ belongs to $\mathcal{SL\text{-}P}$. The result would provide the converse to the paper's theorem relating shifted Laguerre–Pólya functions to eventual positivity of their Laguerre operators.