The escaping-trajectory conjecture for Rauzy-gasket triangle tilings

Let $(x_1,x_2,x_3)$ be a point in the Rauzy gasket, meaning that $x_1+x_2+x_3=1$ and the stated subtraction-and-rescaling algorithm can be applied infinitely many times. Consider the triangle tiling with angles

$$\alpha=\pi(1-x_1)/2,\qquad \beta=\pi(1-x_2)/2,\qquad \gamma=\pi(1-x_3)/2.$$

A trajectory through the circumcenter has parameter $\tau=\pi$.

Rauzy-gasket escaping-trajectory conjecture. Every trajectory through the circumcenter is escaping, and trajectories passing increasingly close to the circumcenter, with $\tau\to\pi^-$, exhibit a fractal structure under rescaling.

The source states that Hubert and Paris-Romaskevich subsequently proved the first assertion and proved that this is the only way to obtain a nonlinearly escaping trajectory. The fractal-rescaling assertion is not resolved in the supplied text.