The one-dimensional Mizohata–Takeuchi conjecture for convex curves

Let $\Gamma$ be a convex $C^2$ function on $[0,1]$, let $E_\Gamma$ be its extension operator, and let $\omega:B_R\to[0,\infty)$ be a one-dimensional weight, meaning

$$\omega(B(y,r))\le r$$

for every ball of radius $r$ centered at $y$, with $1\le r\le R$. One-dimensional Mizohata–Takeuchi conjecture. Then

$$\int_{B_R}|E_\Gamma g|^2\omega\lesssim_\epsilon R^\epsilon\sup_T\omega(T)^{\frac12}\|g\|_2^2,$$

where $T$ ranges over $R\times R^{\frac12}$ tubes. The paper explicitly says that this conjecture is false for general $C^2$ convex curves, so the claim is refuted in that stated generality.