Regarding convergence, make sure you’re indexing your parameters correctly. As I understand your code, each element of the alpha matrix is a parameter estimate of the relationship between X and the mean of some proportion of interest beta_hat.
However you declare only a single variance parameter phi. Unless you have some reason to constrain the variance of your proportion estimates across the 15 groups you plot to be the same, I’d expect the number of phis to be the same as the number of alphas, which should be the same as the number of proportions your modeling. Again it seems like you’re modeling 15 proportions in the top two plots (but in the last figure seems like only 5).