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

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