The local Mizohata-Takeuchi conjecture

Let $\Sigma$ be a $C^2$ hypersurface, let $B_R\subset\mathbb R^n$ be an $R$-ball, and let $w$ be a non-negative weight. With $\operatorname{ext}_{\Sigma}$ the Fourier extension operator and $\mathcal M$ the Kakeya-type maximal operator on $S^{n-1}$, the conjecture asserts:

Local Mizohata-Takeuchi conjecture. For every $\alpha>0$, one has

$$\|\operatorname{ext}_{\Sigma}f\|_{L^2(B_R;w(x)\,dx)}^2\lesssim R^\alpha\|\mathcal M w\|_{L^\infty(S^{n-1})}\,\|f\|_{L^2(\Sigma)}^2.$$

The local conjecture asks whether the estimate can hold with arbitrarily small power loss. The paper's abstract says that it fails with a definite power loss for a dense family of compact $C^k$ convex hypersurfaces, including many $C^2$ hypersurfaces, but the parser supplied no formal resolution status for the conjecture itself.