The lacunary-sided polygon restricted maximal conjecture

Let $P_{\mathrm{lac}}$ be the lacunary-sided polygon in the plane whose first-quadrant vertices are the points $e^{\pi i2^{-k}}$ for $k\in\mathbb N$. For $k\in\mathbb N$, define

$$T_kf(x)=\int_{(1+2^{-k})P_{\mathrm{lac}}}\widehat f(\xi)e^{i\xi\cdot x}\,d\xi.$$

Restricted lacunary-sided polygon conjecture. For $2\le p<\infty$, the maximal operator $f\mapsto\sup_{k\in\mathbb N}|T_kf|$ maps $L^p(\mathbb R^2)$ into itself. The source calls this a restricted version of the preceding polygon-dilation conjecture and indicates that it remains challenging and open.