Hörmander-type sufficient condition for hypoellipticity on p.c.f. fractals

Let $P$ be a polynomial in $m$ variables and let $P(\Delta)$ be the corresponding operator on a product of p.c.f. fractals. For a multi-index $\kappa$, write $D^{\kappa}P$ for the corresponding partial derivative. Hörmander-type hypoellipticity conjecture. A sufficient condition for the hypoellipticity of $P(\Delta)$ is

$$\frac{D^{\kappa}P(\xi)}{P(\xi)}\longrightarrow 0\quad\text{as }\xi\to\infty$$

for every partial derivative $D^{\kappa}$ with $|\kappa|>0$. This is proposed as a fractal analogue of the Euclidean hypoellipticity criterion, but the source does not establish the condition in the fractal setting.