One-dimensionality conjecture for the path variety of a finite union of algebraic curves

Let $M\subset\mathbb{R}^d$ be a finite union of algebraic curves, and let $P_{\in M}$ denote the associated path variety. Path-variety dimension conjecture. Then $P_{\in M}$ is one dimensional. This asserts that the algebraic variety generated by paths in a finite union of algebraic curves has dimension one; the supplied text does not indicate whether the claim has been resolved.