The Fourier-transform conjecture for real transcendental entire functions in the Laguerre–Pólya class

Let $H(t)=\sum_{k=0}^\infty \gamma_k x^k/k!$ be a real transcendental entire function, with $H'\in\mathscr{L{\kern.2ex\text{\raise.5ex\text{{\rule{.35em}{.12ex}}}}\kern.2ex}P}$, and suppose that $H(it)\to -\infty$ as $t\to\pm\infty$. Define

$$F(z)=\int_{-\infty}^\infty e^{H(it)}e^{izt}\,dt.$$

Fourier-transform conjecture. The function $F$ belongs to the Laguerre–Pólya class:

$$F\in\mathscr{L{\kern.2ex\text{\raise.5ex\text{{\rule{.35em}{.12ex}}}}\kern.2ex}P}.$$

This conjecture proposes that the stated decay and Laguerre–Pólya condition on $H'$ force the Fourier transform of $e^{H(it)}$ to remain in the Laguerre–Pólya class. The source presents it as a direction for further investigation; no resolution is supplied there.