On the dichotomy of $p$-walk dimensions on metric measure spaces

On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincaré inequality and the cutoff Sobolev inequality with $p$-walk dimension ${\beta }_p$ hold, for $p$ in an open interval $I\subseteq (1,+\mathrm{\infty })$, we prove the following dichotomy: either ${\beta }_p=p$ for all $p\in I$, or ${\beta }_p>p$ for all $p\in I$.