Invariant-property characterization of $\rho$-irregularity

Let $(\gamma,\rho)$ be a pair of parameters. A continuous path is a function $f\in C^0_t$, and a time reparametrization is composition with the inverse of a homeomorphism $\tau$. A path $g$ is $(\gamma,\rho)$-irregular when it has the $(\gamma,\rho)$-irregularity property defined earlier in the paper.

Invariant-property conjecture. For any pair $(\gamma,\rho)$, there exists a property $\mathcal{P}$ such that:

1. For any $f\in C^0_t$ with property $\mathcal{P}$, there exists a homeomorphism $\tau$ such that $g=f\circ\tau^{-1}$ is $(\gamma,\rho)$-irregular.
2. The property $\mathcal{P}$ is invariant under time reparametrization.

The conjecture seeks an intrinsic characterization of paths that can be made $(\gamma,\rho)$-irregular by time change, analogous to the invariance of finite variation under reparametrization. The paper does not establish the existence or identify the property $\mathcal{P}$, so the conjecture remains open.