Sharp bandwidth exponent conjecture for surfaces of revolution

Let $\mathcal{S}\in\mathfrak{S}$, let $\varepsilon>0$, and let $K$ be a compact subset of $\mathcal{S}$ at distance at least $\varepsilon$ from both the poles and the equator. Let $P_{\lambda,\delta}$ be the spectral projector in the paper.

Sharp bandwidth conjecture. The main projector estimate should hold for

$$\delta\geq\lambda^{-1/3},\qquad \\|P_{\lambda,\delta}\\|_{L^2(\mathcal{S})\to L^\infty(K)}\lesssim_{\mathcal{S},\varepsilon}\lambda^{1/2}\delta^{1/2}.$$

Furthermore, the lower bound of the cited corollary should also hold up to $\lambda^{-1/3}$. For any $\mathcal{S}$, the exponent $1/3$ is conjectured to be optimal: the main estimate should fail at scales $\lambda^{-\tau}$ for every $\tau>1/3$. This is the paper's conjectural optimal improvement of the proved exponent $1/32$, away from the poles and equator.