The Curve Histogram Conjecture for fully regular plane curves

Let $\mathcal{X}$ be a metric measure space whose underlying space $X$ is a plane curve, equipped with extrinsic Euclidean distance and normalized arclength measure. A fully regular plane curve is a plane curve in the regularity class considered by Brinkman and Olver, including polygons and generic smooth curves. Let $H_{\mathcal{X}}$ denote its distance histogram.

Curve Histogram Conjecture. The distance histogram $H_{\mathcal{X}}$ determines a fully regular plane curve up to isometry.

This conjecture asks whether global distance distributions distinguish fully regular plane curves. Brinkman and Olver established convergence results for discrete approximations of plane curves, but the stated reconstruction claim remains open in the supplied source.