The rational-angle conjecture for flares in the parameter plane

Let the centered unit disk in the parameter plane be the disk centered at $0$ of radius $1$, and let a flare be one of the sun-flare-like parameter sets arising in the coded-coloring of the parameter plane. Rational-angle flare conjecture. Flares in the parameter plane meet the centered unit disk at points whose argument is a rational multiple of $\pi$. The source motivates this assertion by experimental observations that the central curves of flares appear to meet the unit circle at such points; no resolution is given.