Curvature-controlled Fourier decay conjecture for convex planar sets

Let $K\subset\mathbb{R}^2$ be a convex compact set with piece-wise smooth boundary, and let $\kappa(P)$ denote the curvature at $P\in\partial K$. Set

$$\nu=\min_{P\in\partial K}\kappa(P).$$

Curvature-controlled Fourier decay conjecture. There exists an absolute constant $C_1>0$ such that

$$\sup_{\omega\in\mathbb{R}^2}|\omega|^{3/2}|\widehat{\chi}_K(\omega)|\le\frac{C_1}{\sqrt{\nu}}.$$

Moreover, this estimate is sharp: there exists an absolute constant $C_2>0$ and a sequence $\{\omega_n\}\subset\mathbb{R}^2$ with $|\omega_n|\rightarrow\infty$ such that

$$|\omega_n|^{3/2}|\widehat{\chi}_{K}(\omega_n)|\ge\frac{C_2}{\sqrt{\nu}}+o(1).$$

The conjecture proposes a curvature-dependent uniform Fourier decay estimate for the indicator of a convex planar set, together with sharpness of the dependence on the minimum curvature. The supplied text does not establish the claim or provide evidence of its resolution, so its status remains open.