Stein's conjecture for weighted extension operators

Let $\Sigma\subset\mathbb{R}^d$ be a $C^2$ hypersurface with surface measure $\mathrm{d}\sigma$, let $f\in L^2(\Sigma;\mathrm{d}\sigma)$, and let $w$ be a nonnegative weight. Let $N(\sigma)$ denote the normal to $\Sigma$ at $\sigma$, and let $Xw$ be the X-Ray transform of $w$. Stein's conjecture. Under the hypotheses of the Mizohata–Takeuchi conjecture, one has

$$\int_{\mathbb{R}^d}|\mathcal E f|^2w\,\mathrm{d}x\lesssim\int_\Sigma |f(\sigma)|^2\sup_{\ell\parallel N(\sigma)}Xw(\ell)\,\mathrm{d}\sigma(\sigma).$$

The supplied text presents this as a conjecture arising from the connection between Kakeya or Nikodym maximal functions and Bochner–Riesz multipliers; it gives no resolution status.