Iwaniec–Sarnak local $L^p$-norm conjecture for Hecke–Maass forms

Let $\phi$ be a Hecke–Maass form for the full modular group, let $K$ be a compact subset of the modular surface $\mathbb{X}$, let $2<p\le\infty$, and let $\epsilon>0$. Then Iwaniec–Sarnak's local $L^p$-norm conjecture. There exists $c=c(p,K,\epsilon)$ such that

$$\left(\int_K\left|\phi(z)\right|^p\frac{dxdy}{y^2}\right)^{\frac{1}{p}}\le c\lambda_{\phi}^{\epsilon}\left(\int_K\left|\phi(z)\right|^2\right)^{\frac{1}{2}}.$$

This is a local sup-norm and moment conjecture proposed by Iwaniec and Sarnak; the source gives no resolution status. The displayed formulation uses the local $L^2$ quantity exactly as stated in the paper.