Chains differ in categorical model

Are both beta and gamma truly centered over the same value? In the simulation both are centered over 0 with a sd of 1. There’s an identifiability issue. The model could easily find one value with a mean of 0 and a sd of sqrt(2), which is what you get by adding the two rv together. If you simulate with distinct means that are far enough away in sd land then what does the inference look like?