Preserve one, preserve all

Isometries of metric spaces $(X,d)$ preserve all level sets of $d$. We formulate and prove cases of a conjecture asserting if $X$ is a complete Riemannian manifold, then a function $f:X\to X$ preserving at least one level set $d^{-1}(r)$, with $r>0$ small enough, is an isometry.