Conjecture on singular points in trajectories of planar fractional-order systems

Let $0<\alpha\leq 1$ and consider the planar fractional-order system

$${}_0^C\mathrm{D}_t^\alpha X(t)=AX(t),$$

where $A$ has eigenvalues $\lambda=re^{\pm i\theta}$. Singular points are points at which the trajectory is not smooth in a neighborhood.

Singular-point conjecture. There exist singular points in the trajectory if and only if

$$\frac{\alpha\pi}{2}-\delta_1 < |\operatorname{arg}(\lambda)| < \frac{\alpha\pi}{2}+\delta_2,$$

where $\delta_1>0$ and $\delta_2>0$ are sufficiently small positive real numbers.

The conjecture is based on observed self-intersections and cusps in trajectories of autonomous planar fractional-order systems. The source presents it as an observation-based conjecture; no proof or resolution is given.