The maximal function conjecture for the two-dimensional Schrödinger equation

Let $A(1)=\{\xi:|\xi|\sim 1\}$. For $R\geq 1$, let $B(0,R)$ denote the ball of radius $R$ centered at the origin, and let $e^{it\Delta}f$ be the free Schrödinger evolution of a function $f\in L^2(\mathbb R^2)$ whose Fourier transform is supported in $A(1)$. Maximal function conjecture. For every $p\geq 3$ and every $\epsilon>0$, there is a constant $C_\epsilon$ such that, for all $R\geq 1$ and all such $f$,

$$\left\|\sup_{0<t\leq R}|e^{it\Delta}f|\right\|_{L^p(B(0,R))}\leq C_\epsilon R^\epsilon\|f\|_2.$$

This estimate is the scale-invariant maximal-function bound expected to imply the relevant pointwise convergence result for the two-dimensional Schrödinger equation; its status is not resolved in the supplied text.