This is motivated by this older question. Now posted at MO.
Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more interesting question) equipped with the Hausdorff metric. Every finite set of contraction maps $F=\{f_1,...,f_n\}$ on $\mathbb{R}^2$ yields a single contraction map on $\mathscr{H}$, namely $$\hat{F}: X\mapsto \bigcup_{1\le i\le n}f_i[X].$$ Since $\mathscr{H}$ is complete, by Banach $\hat{F}$ has a unique fixed point $S_F$. Let $\mathscr{F}$ be the subspace of $\mathscr{H}$ consisting of non-singleton sets of the form $S_F$ for some $F$ as above.
Now for $A,B\in\mathscr{F}$, write $A\perp B$ iff there are finite sets of contraction maps $F,G$ on $\mathbb{R}^2$ and a nonempty compact set $X\subseteq\mathbb{R}^2$ with $dim(X)=2$ such that $$A=S_F, B=S_G,F[X]\cup G[X]=X,\mbox{ and }dim(F[X]\cap G[X])<2.$$ (The intersection-dimension condition basically says that $F[X]$ and $G[X]$ are "as disjoint as they could be" - we generally won't be able to avoid overlap on boundaries, but we can ask that there not be any "substantial" overlapping.) From $\perp$ we get an equivalence relation generated by "shared complements" - let $A\sim B$ iff there is an even-length sequence $C_1,C_2,...,C_{2k-1}, C_{2k}\in\mathscr{F}$ with $C_1=A$, $C_{2k}=B$, and $C_i\perp C_{i+1}$ for each $1\le i<2k$. My question is:
How many classes does $\sim$ partition $\mathscr{F}$ into?
"Obviously" the answer is continuum-many ... right? At the moment, though, I can't even show that the answer is $>1$!