Whether $p$-conductive homogeneity holds depends on $p$

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