Gibbs Priors form a Martingale

Question

I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as a martingale. Specifically, I am trying to form a sequence of $q^*_n$'s where $$q^*_n(\theta) = \frac{\exp(-L(\theta, x)) \pi(\theta)}{\int \exp(-L(\theta, x)) \pi(\theta) \, d\theta}$$ such that it forms a martingale (either normal, sub or super martingale), as new data is imputed. Any ideas?