Problem with beta distribution

I’m trying to simulate the posterior distributions for almost 13000 parameters. I’ve run the model with as many as 15000 iterations per chain (4 chains). But I’m getting warning messages related to low bulk and tail ESS as well as low BFMI. When trying to find out what’s wrong, I decided to assign arbitrary values to the hyperparameters of the beta priors for the parameters p[K,N] you can see below. Then I ran the model on a small subset of my original sample. It turns out that when I assigned values close to 0 to the alpha parameter of the beta prior, say 0.01, I got those same warning messages. But when I raised alpha to a higher value that is still less than 1, say 0.3, the simulation ran smoothly with no warning messages. I’m guessing this is what is causing the problem in my model since the hyperparameters alpha[K] below are quite close to 0. Do you have any suggestions for some kind of reparametrization? Thank you.

functions {
  real ll_a_lpdf(real p, real d){
    return d*log1m(p);
  }
  real ll_b_lpdf(real p, real d){
    return log1m(pow((1-p),d));
  }
}
data {
  int K; 
  int N;
  int x[K,N];
  real m[K];
}
parameters {
  real <lower=0> d[N];
  real <lower=3,upper=8> mu;
  real <lower=1./4,upper=2> sigma;
  real <lower=0,upper=1> p[K,N];
  real <lower=0,upper=1> rho[K];
}
transformed parameters {
  real <lower=0> alpha[K];
  real <lower=0> beta[K];
  for (k in 1:K){
    alpha[k] = (m[k]*((1/rho[k])-1));
  }
  for (k in 1:K){
    beta[k] = ((1-m[k])*((1/rho[k])-1));
  }
}
model {
  //Prior
  d ~ lognormal(mu,sigma);
  mu ~ uniform(3,8);
  sigma ~ uniform(1./4,2);
  rho ~ uniform(0,1);
  for (i in 1:N){
    for (k in 1:K){
      p[k,i] ~ beta(alpha[k],beta[k]);
    }
  }
  // Likelihood
  for (i in 1:N){
    for (k in 1:K) {
      if (x[k,i]==0) {
        target += ll_a_lpdf(p[k,i]|d[i]);
      } else {
        target += ll_b_lpdf(p[k,i]|d[i]);
      }
    }
  }
}