If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question?
Again, the new local ring $R$ is of zero Krull dimension, and any ideal generated by a finite number of powers of $\bar X_i$'s is nilpotent.
Thanks for any cooperation!