SCHWEITZER 2020 Problem 6

Does there exist an entire function $F \colon \mathbb{C}\to \mathbb{C}$ such that $F$ is not zero everywhere, $|F(z)|\leq e^{|z|}$ for all $z\in \mathbb{C}$ , $|F(iy)|\leq 1$ for all $y\in \mathbb{R}$ , and $F$ has infinitely many real roots.