@mei@donotsta.re He discusses this in detail in section 8 of the paper (and it's included in a branch of https://github.com/digama0/ConZF), but accessibility is not stable, and he quotes me as showing that the natural accessibility hypothesis for stable membership implies the double negation of Markov's Principle (and the accessibility hypothesis itself can be viewed as a transfinite form of Markov's Principle).