I wonder if there is a way to demonstrate that the problem is multimodality rather than other factors that would cause the parameters to get stuck. Perhaps the log likelihood distribution for each chain would be similar if they land on different modes. If you are getting stuck in areas with low posterior probability, the chains would have different log likelihood distributions. I think this is the code you would need:
histogram(chains[:log_density], alpha=.5, legend=:outerright)