Non-spectrality conjecture for piecewise smooth closed curves

Let $mu$ be the arc-length measure on a piecewise smooth closed curve in the plane.

Closed-curve non-spectrality conjecture. The measure $mu$ is not spectral.

The paper has already proved non-spectrality for several classes of piecewise smooth curves, including closed polygonal lines, but does not classify all piecewise smooth curves. The conjecture asks whether every piecewise smooth closed curve is non-spectral.