The difference-of-roots conjecture for the Enriques lattice

Let $\Lambda=U\oplus \mathbb{E}_8(-1)$ be the even unimodular lattice of signature $(1,9)$, identified with the cohomology lattice of an Enriques surface up to torsion. Difference-of-roots conjecture. Every element $H\in\Lambda$ can be written as

$$H=D_1-D_2,$$

where $D_1,D_2\in\Lambda$ satisfy

$$D_1^2=D_2^2=-2.$$

This purely lattice-theoretic assertion would imply the existence of the required numerical classes for Ulrich line bundles on unnodal Enriques surfaces, but the source states that it is not presently proved.