Counterexamples for parameter values other than p=q=1/2

Determine whether there exists a counterexample, for values of p and/or q other than p=q=1/2, to the conjecture that if an event A has constant conditional probability q given every finite Bernoulli-pattern event B^{S,T}, then A is independent of the entire sequence (B_i)_{i\in\mathbb{N}}.

Background

The paper studies independent events B_1,B_2,\ldots, each having probability p, and an event A whose conditional probability given finite intersections of events B_i and their complements is q. Corollary 1 establishes that, under the relevant equi-dependence condition for fixed nonnegative integers s and t, A is independent of each finite-pattern event B^{S,T}.

The authors conjecture that this finite-pattern independence might extend to independence from the entire sequence of events. Example 1 disproves that conjecture when p=q=1/2 by taking A to be the parity event D_n determined by the first n Bernoulli trials. The unresolved issue is whether analogous counterexamples exist for other choices of p and/or q.