  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 $\sqrt{6}/80$-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.
