Lu–Zhou conjecture on the $L^p$-essential spectrum

Let $M$ be a complete, non-compact Riemannian manifold with Ricci curvature bounded below and uniformly sub-exponential volume growth. For $p\geq 1$, the $L^p$-essential spectrum is the essential spectrum associated with the Laplacian acting on $L^p(M)$. Lu–Zhou conjecture. For every $p\geq 1$, the $L^p$-essential spectrum is

$$[0,\infty).$$

The conjecture concerns the spectral behavior of complete manifolds without asymptotic nonnegativity of Ricci curvature. The paper states that its result provides counterexamples, so the conjecture is disproved.