Thank you, Bob, for the kind comment.
According the Wikipedia you mentioned, “Similarly, the Jeffreys prior for \ln\sigma^2=2\ln\sigma is also uniform”. So, I don’t think the mathematical expression \pi(\mu,\sigma_1^2,\dotsb,\sigma_C^2)\propto\prod_{i=1}^C\sigma_i^{-2} is incorrect. The complete code and data can be found on my previous post in the Stan forum here.