Källén–Lehmann spectrum conjecture for propagators on hyperbolic and Euclidean manifolds

Let $M$ be a manifold whose universal cover is $\mathbb{H}^{d}$ or $\mathbb{R}^{d}$. In the hyperboloid model, or in the pacman model, consider the proper times associated with the deck transformations of a point $y$, and suppose these proper times can be partitioned into disjoint periodic collections of the form $\alpha_i\mathbb{N}+b_i$. Denote the resulting partition by $W_M$.

Källén–Lehmann spectrum conjecture. The free scalar propagator on $M$ has a Källén–Lehmann power spectrum $\rho(m)$ equivalent to

$$\rho(m)=\delta(m)+\sum_{i=1}^{\infty}\frac{a_i\delta(m-m_i)}{m},$$

where $\\{m_i\\}_{i=1}^{\infty}\subset\mathbb{R}$ is a discrete set whose only cluster point is zero, with $m_i\to0$ implying $a_i\to0$. For each $m_i$, there is some $\alpha_i\in W_M$ such that $\alpha_i m_i\in4\pi\mathbb{N}$.

The source explicitly says that this is a theorem for manifolds whose universal cover is $\mathbb{R}^{d}$; it does not provide a resolution status for the corresponding claim in the broader hyperbolic-or-Euclidean formulation.