The planar $L^{3/2}$ maximal-operator conjecture

For $t\in[1,2]$ and $R\geq1$, let $S_{t,R}$ be the Fourier multiplier operator on $\mathbb R^2$ defined by

$$\widehat{S_{t,R}f}(\xi)=a(R(t-|\xi|))\widehat f(\xi),$$

where $a$ is a standard bump function supported on $[-1,1]$, and define

$$S_R^*f(x):=\sup_{t\in[1,2]}|S_{t,R}f(x)|.$$

The planar $L^{3/2}$ conjecture. The estimate

$$\big\|S_R^*f\big\|_{L^p(\mathbb R^2)}\leq C_\varepsilon R^\varepsilon\|f\|_{L^p(\mathbb R^2)}$$

holds for $p=3/2$, every $\varepsilon>0$, every Schwartz function $f$, and every $R\geq1$. This is identified as the conjectural estimate that would settle Tao's conjecture in the planar case; the supplied text does not state whether it is resolved.