Cantor-set conjecture for orderings of free groups

Let $F_n$ be the free group on $n>1$ generators. Write $LO(F_n)$ for the space of left-orderings of $F_n$ and $BiO(F_n)$ for the space of bi-orderings, each with the topology inherited from the space of orderings. Cantor-set conjecture. The spaces $LO(F_n)$ and $BiO(F_n)$ are homeomorphic to the Cantor set. This predicts that both ordering spaces have the same standard compact, perfect, totally disconnected topology; the supplied text gives no resolution of the claim.