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Secret sharing is an essential tool for many distributed applications, including distributed key generation and multiparty computation. For many practical applications, we would like to tolerate network churn, meaning participants can dynamically enter and leave the pool of protocol participants as they please. Such protocols, called Dynamic-committee Proactive Secret Sharing (DPSS) have recently been studied; however, existing DPSS protocols do not gracefully handle faults: the presence of even one unexpectedly slow node can often slow down the whole protocol by a factor of O(n). In this work, we explore optimally fault-tolerant asynchronous DPSS that is not slowed down by crash faults and even handles byzantine faults while maintaining the same performance. We first introduce the first high-threshold DPSS, which offers favorable characteristics relative to prior non-synchronous works in the presence of faults while simultaneously supporting higher privacy thresholds. We then batch-amortize this scheme along with a parallel non-high-threshold scheme which achieves optimal bandwidth characteristics. We implement our schemes and demonstrate that they can compete with prior work in best-case performance while outperforming it in non-optimal settings.