Exponential wave trace remainder conjecture for asymptotically hyperbolic manifolds

Let $(X,g)$ be an asymptotically hyperbolic $(n+1)$-dimensional manifold with negative sectional curvatures. Write

$$for the Laplacian,$$

for the primitive closed geodesics, and

$$for the linearized Poincaré map along$$

. As a distributional equality in $\mathcal{D}'((0,\infty))$,

$$\operatorname{0\text{-}tr}\cos\left(t\sqrt{\Delta-n^2/4}\right)=\sum_{\gamma\in\mathcal{L}_p}\sum_{k=1}^{\infty}\frac{l(\gamma)\delta(t-kl(\gamma))}{\sqrt{|\det(I-\mathcal{P}_\gamma^k)|}}+A(t).$$

Exponential wave trace remainder conjecture. The remainder $A$ is smooth on $(0,\infty)$, and there exist $C,k>0$, depending only on $n$, $k_1$, and $k_2$, such that

$$|A(t)|\leq Ce^{kt}\qquad\forall t>1,$$

whenever the sectional curvatures $\kappa$ satisfy $-k_1^2\leq\kappa\leq-k_2^2$.

This conjecture seeks a controlled exponential remainder in the wave trace formula under pinched negative curvature. The supplied text presents it as an expectation, and no resolution is given here.