Weak-type spherical Carleson conjecture

Let $d>1$. For $f\th?$, define the spherical Carleson operator by

$$C_Sf(x)=\sup_{R>0}\left|\int_{|\xi|\leq R}\widehat f(\xi)e^{2\pi i\xi\cdot x}\,d\xi\right|.$$

Spherical Carleson conjecture. The estimate

$$\|C_Sf\|_{L^{2,\infty}(\mathbb{R}^d)}\lesssim\|f\|_{L^2(\mathbb{R}^d)}$$

should hold for every $f\in L^2(\mathbb{R}^d)$. This is the remaining endpoint case highlighted after the failure of strong $L^p$ boundedness for $p\ne2$; the supplied text gives no resolution.