  We introduce two fractals, in Euclidean spaces of dimension two and three
respectively, such the $2$-conductive homogeneity holds but there is some $\eps
\in (0, 1)$ so that the $p$-conductive homogeneity fails for every $p\in (1,
1+\eps)$. In addition, these two fractals have Ahlfors regular conformal
dimension within the interval $(1, 2)$ and $(2, 3)$, respectively.
