Ellipticity implies hypoellipticity for operators on p.c.f. fractals

Let $X$ be a p.c.f. fractal, let $P$ be a polynomial, and let $P(\Delta)$ denote the associated operator on the product space $X^m$. The operator $P(\Delta)$ is elliptic when $P(\xi_1^2,\dotsc,\xi_m^2)$ is elliptic, and it is hypoelliptic when

$$\operatorname{SingSppt}\bigl(P(\Delta)T\bigr)=\operatorname{SingSppt}(T)$$

for every distribution $T\in\mathcal{D'}(\Omega)$. Ellipticity-hypoellipticity conjecture. If $X$ is a p.c.f. fractal and $P(\Delta)$ is an elliptic operator on the product space $X^m$, then $P(\Delta)$ is hypoelliptic. For $m=1$, all operators $P(\Delta)$ are elliptic and are known to be hypoelliptic; the conjecture concerns products of p.c.f. fractals and is motivated by the corresponding theory for elliptic operators in Euclidean analysis.