Biased-normal characterization of attractor-generating sequences

Let $(X,d)$ be a compact metric space, and let $\{X;w_0,w_1,\dotsc,w_n;p_0,p_1,\dotsc,p_{n-1}\}$ be a hyperbolic iterated function system (IFS) with probabilities. Let $\nu$ be its invariant measure, whose support is the IFS attractor. For a sequence $\sigma$ used by the determined iteration algorithm, let $S$ be the set of sequences for which every Borel subset $B\subseteq X$ with null boundary is visited with limiting frequency $\nu(B)$.

Attractor-generation conjecture. Given a hyperbolic IFS with probabilities, a sequence $\sigma$ is an element of $S$—that is, $\sigma$ generates the attractor of the IFS as described above—if and only if $\sigma$ is biased normal with respect to the probabilities of the IFS.

The conjecture seeks an exact characterization of the sequences that reproduce the invariant-measure frequencies, and hence the attractor, under determined iteration. The surrounding text records that sequences with the matching Bernoulli measure belong to $S$ almost surely, but gives no proof of the proposed if-and-only-if characterization.