The characterization of locally interval graphs as circular-arc graphs

Given a graph $G$ and a vertex $v$ of $G$, let $B_G(v,r/2)$ be the subgraph induced by vertices at distance at most $\lfloor r/2\rfloor$ from $v$, with edges $xy$ satisfying $d(v,x)+d(v,y)<r$. The graph $G$ is $r$-locally interval if every such ball is an interval graph. A graph is circular-arc if it has a region representation over a cycle in which every region is a path.

Rz\k{a}.{z}ewski's conjecture. Locally interval graphs are exactly circular-arc graphs.

This conjecture proposes that being locally path-like, in the sense that all sufficiently specified local balls are interval graphs, is equivalent to admitting a region representation over a cycle. The source gives no resolution status.