The Reuleaux-frame conjecture for closed equilateral polygonal chains

Let $C$ be a closed equilateral polygonal chain in the Euclidean plane, and let its diameter be the supremum of distances between points of $C$. Reuleaux-frame conjecture. If the diameter of $C$ is equal to the length of a line segment of $C$, then $C$ is the frame of some Reuleaux polygon. The paper uses this unproved claim in a proof sketch concerning extremal diameter configurations in the plane; no proof or disproof is supplied in the given text.