Existence and Hadamard property of states of low energy

Let $f\in C_0^\infty(I)$ be a smearing function, where $I\subset\mathbb{R}$ is the domain of conformal time, and consider a scalar field of mass $m$ on a cosmological spacetime with scale factor $a$ and curvature coupling $\xi$. States of low energy are obtained by minimizing the smeared energy density per mode. Existence and Hadamard conjecture. States of low energy for arbitrary smearing function, mass and scale factor exist only when

$$0\leq\xi\leq\frac{1}{6}.$$

For every such $\xi$, the state satisfies the microlocal spectrum condition. This condition is equivalent to the state being Hadamard, so the conjecture gives both the allowed curvature-coupling range for existence and the expected regularity of the resulting states. The preceding discussion establishes these properties for minimally and conformally coupled scalar fields, but the assertion for arbitrary curvature coupling in the stated range remains conjectural.