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Guerraoui proposed an indulgent solution for the binary consensus problem. Namely, he showed that an arbitrary behavior of the failure detector never violates safety requirements even if it compromises liveness. Consensus implementations are often used in a repeated manner. Dutta and Guerraoui proposed a zero-degrading solution, \ie during system runs in which the failure detector behaves perfectly, a node failure during one consensus instance has no impact on the performance of future instances. Our study, which focuses on indulgent zero-degrading binary consensus, aims at the design of an even more robust communication abstraction. We do so through the lenses of self-stabilization - a very strong notion of fault-tolerance. In addition to node and communication failures, self-stabilizing algorithms can recover after the occurrence of arbitrary transient faults; these faults represent any violation of the assumptions according to which the system was designed to operate (as long as the algorithm code stays intact). This work proposes the first, to the best of our knowledge, self-stabilizing algorithm for indulgent zero-degrading binary consensus for time-free message-passing systems prone to detectable process failures. The proposed algorithm has an O(1) stabilization time (in terms of asynchronous cycles) from arbitrary transient faults. Since the proposed solution uses an Ω failure detector, we also present the first, to the best of our knowledge, self-stabilizing asynchronous Ω failure detector, which is a variation on the one by Mostéfaoui, Mourgaya, and Raynal.