Pointwise Weyl-law improvement for selected surfaces of revolution

Let $\mathcal{S}\in\mathfrak{S}$, and let $K_\varepsilon$ be the set of points of $\mathcal{S}$ at distance at least $\varepsilon$ from both poles. Let $N(\lambda,x)$ denote the pointwise eigenvalue counting function and let $c_W(x)$ be the local Weyl coefficient.

Pointwise Weyl-law improvement conjecture. There exists a quantifiable constant $\kappa>0$ such that, for every $x\in K_\varepsilon$,

$$N(\lambda,x)=(2\pi)^{-2}c_W(x)\lambda^2+O_{\mathcal{S},\varepsilon}(\lambda^{1-\kappa}).$$

This predicts a polynomial improvement over the usual pointwise Weyl-law remainder on the specified surfaces, uniformly away from the poles. The authors state it as an expected consequence of their methods; no resolution is supplied.