Strichartz remainder conjecture for compact Heisenberg manifolds

Determine whether, for every compact Heisenberg manifold M=Γ\H_d, the spectral counting remainder R_M(λ)=N_M(λ)−A_d vol(M)λ^{d+1} satisfies R_M(λ)=O_M(λ^d).

Background

For the standard sub-Laplacian on a compact Heisenberg manifold M=Γ\H_d, Strichartz had previously established the Weyl law with remainder O_M(λ^d log λ). The conjecture proposed that the logarithmic factor could be removed, yielding the optimal polynomial remainder order λ^d.

The paper subsequently states that its two-sided lower bound R_M(λ)=Ω_{M,±}(λ^d log log λ) disproves this conjecture. Thus the conjecture is explicitly recorded in the paper but is resolved negatively by the paper's main results.