To have something under your belt means to have safely or satisfactorily achieved it, and in this context sobriety is something that needs to be achieved in order to show commitment.
Context: "We won't do it until you have a year of sobriety to show that you're committed, to make absolutely sure that you're on stable ground." Claim: "We won't do it until you have a year of sobriety under your belt, to make absolutely sure that you're on stable ground."
Considering the context, is the claim true, false, or unknown?