Consider the de-Bruijn-Newman function
$$H_{t}(z)=\int_{0}^{\infty}e^{tu^2}\phi(u)\cos (zu) \mathrm{du},$$ where $\phi$ is a certain super exponentially decaying function. Much work has been done on the function $H$, including in the Polymath15 project.
My question: Let $z=x+iy$. Is there a known $y_0$ such that $H_{t}(x+iy)\neq 0$ for ALL $t<0$ and $|y|\geq y_0$ ?