Metric-independence conjecture for delocalized $L^2$-invariants of hyperbolic manifolds

Let $M$ be a closed odd-dimensional manifold admitting a hyperbolic structure, and let $g$ be any Riemannian metric on $M$. The delocalized $L^2$-invariants of $M$ are defined using the heat kernel on the universal cover $\widetilde{M}$.

Metric-independence conjecture. For every such metric $g$, the delocalized $L^2$-invariants are well-defined and independent of the choice of $g$.

The conjecture would establish that these invariants are genuine topological invariants in the hyperbolic case and support their relation to the marked length spectrum. The paper proves well-definedness and metric independence in several special cases, but leaves the general hyperbolic-manifold statement open.