Newman's conjecture for the De Bruijn–Newman constant

Let $\Xi_t(x)$ be the deformed Riemann zeta function, and let $\Lambda\in\mathbb R$ be the De Bruijn–Newman constant, characterized by the fact that $\Xi_t$ has only real zeros for $t\geq\Lambda$ and has a non-real zero for $t<\Lambda$. Newman's conjecture. One has

$$\Lambda\geq 0.$$

The Riemann hypothesis is equivalent to $\Lambda\leq 0$, so the conjecture asserts that the Riemann hypothesis, if true, is only barely true. The source does not state a resolution of this conjecture.