Lack to fit in bayesian beta regression model

I have a question about beta regression model.
I fitted a bayesian beta regression.

- Coefficients are probabilistically (using the posterior distributions) different of zero.
- Gelman-Rubin statistics (for each coefficient) are very close to 1.

But, the bayesian p-value is very far of 1/2. Therefore, there are lack to fit of the data to the model. Do you know any alternative to model a bounded variable in function of several independent variables?
Thanks a lot for your help and suggestions!

Kind regards,