Finiteness conjecture for covering spectra of Laplace-isospectral manifolds

Let $\mathcal{M}$ be a set of Laplace-isospectral manifolds with a uniform upper bound on diameter. Finiteness conjecture. There are only finitely many distinct covering spectra for the manifolds in $\mathcal{M}$. The known result in the paper establishes this when the manifolds are negatively curved; the conjecture removes that curvature assumption while retaining the diameter bound.