Sharp Fourier extension for functions with localized support on the circle

A well known conjecture states that constant functions are extremizers of the $L^2\to L^6$ Tomas-Stein extension inequality for the circle. We prove that functions supported in a $\sqrt{6}\mathrm{/}80$-neighbourhood of a pair of antipodal points on $S^1$ satisfy the conjectured sharp inequality. In the process, we make progress on a programm formulated in arXiv:1509.06674 to prove the sharp inequality for all functions.