Edrei's conjecture on weak local contractions

Let $X$ be a compact metric space, and let $f:X\to X$ be a weak local contraction, meaning that for every $x\in X$ there exists an $r_x>0$ such that $d(x,y)<r_x$ implies $d(f(x),f(y))\leq d(x,y)$. A map $f$ is a local isometry if for every $x\in X$ there exists an $R_x>0$ such that $d(x,y)<R_x$ implies $d(f(x),f(y))=d(x,y)$. Edrei's conjecture. Every weak local contraction $f:X\to X$ on a compact metric space is a local isometry. Edrei's conjecture was disproved by Williams in 1954, who constructed four maps for which every point is a weak contraction point but some points are not isometry points.