Quasiellipticity implies hypoellipticity on p.c.f. fractals

Let $P(\Delta)$ be an operator on a product of p.c.f. fractals, and let $P_0$ be the principal part of $P$. The operator is quasielliptic if there is a constant $c>0$ such that

$$|P_0(\xi)|\geq c|\xi|$$

for every $\xi=(\lambda_1,\dotsc,\lambda_m)$ whose coordinates are eigenvalues of the corresponding Laplacians. Quasiellipticity conjecture. The quasielliptic operators of the cited work are hypoelliptic. The conjecture is motivated by the fact that quasielliptic operators satisfy elliptic-type estimates on the spectrum, although the source explains that quasiellipticity need not coincide with ordinary ellipticity on fractals.