Uniform bounds on rate of merging for Bayesian learners

Update. Cross posted at Cross Validated.

In a well-known paper, Blackwell & Dubins (1962) show that the posterior probabilities of two Bayesian agents, whose priors agree on events of measure $0$, will become arbitrarily close to each other under an increasing stream of information.

Mathematically, the result is as follows. Let $(Omega, mathcal{F}, {mathcal{F}_n}, Q)$ be a filtered probability space with $mathcal{F}_n uparrow mathcal{F}$. Let $P$ be a probability on $(Omega, mathcal{F})$ with $Q ll P$. Then,
$$d(P^n, Q^n): = sup_{A in mathcal{F}}|P(A mid mathcal{F}_n) – Q(A mid mathcal{F}_n)| to 0 text{ a.s. $Q$ as } n to infty.$$
We say that $P$ and $Q$ strongly merge.

In a more recent and also very influential paper, Kalai & Lehrer (1994) introduce the notion of weak merging. The definition is as above, except the $sup$ is taken over finite horizon events; tail events are ignored:
$$w(P^n, Q^n) : = sup_{A in mathcal{F}_{n+1}}|P(A mid mathcal{F}_n) – Q(A mid mathcal{F}_n)| to 0 text{ a.s. $Q$ as } n to infty.$$

For weak merging it is possible to find uniform bounds on the rate of convergence (Fudenberg & Levine, 1992; Sorin, 1999). I am wondering if there are any results in this direction for strong merging.