Brezis's refined topological-degree estimate for circle maps

Let $f ice C^0({\mathbb S}^1;{\mathbb S}^1)$ be a continuous map, and let $\deg f$ denote its topological degree. There should exist constants $C>0$ and $\delta_0>0$ such that, for every $\delta<\delta_0$,

$$|\deg f| \leq C\delta \iint_{|f(s)-f(t)|>\delta} \frac{ds\,dt}{|s-t|^2}.$$

Brezis's conjecture. There exist constants $C>0$ and $\delta_0>0$ such that the displayed inequality holds for every continuous $f\in C^0({\mathbb S}^1;{\mathbb S}^1)$ and every $\delta<\delta_0$. This question asks for a bound on the topological degree of a circle map in terms of a nonlocal integral; the paper proves an inequality inspired by this conjecture, while the conjectured estimate itself is presented as the motivating open problem.