The epsilon-loss local Mizohata–Takeuchi conjecture

Let $B_R$ be the ball of radius $R$ centered at the origin, let $\mathbb{T}$ be the relevant family of tubes, and write $w(T)$ for the weight of a tube. Under the hypotheses of the Mizohata–Takeuchi conjecture, local Mizohata–Takeuchi conjecture. For every $\epsilon>0$,

$$\int_{B_R}|Eg(x)|^2w(x)\,dx\lesssim_{\epsilon}R^{\epsilon}\sup_{T\in\mathbb{T}}w(T)\int_{\Sigma}|g(\omega)|^2\,d\sigma(\omega).$$

This is the localized epsilon-loss version proposed after a counterexample showed logarithmic failure of the scale-invariant local estimate. The source explicitly states that this conjecture remains open.