  Mean-based reconstruction is a fundamental, natural approach to worst-case
trace reconstruction over channels with synchronization errors. It is known
that $\exp(O(n^{1/3}))$ traces are necessary and sufficient for mean-based
worst-case trace reconstruction over the deletion channel, and this result was
also extended to certain channels combining deletions and geometric insertions
of uniformly random bits. In this work, we use a simple extension of the
original complex-analytic approach to show that these results are examples of a
much more general phenomenon: $\exp(O(n^{1/3}))$ traces suffice for mean-based
worst-case trace reconstruction over any memoryless channel that maps each
input bit to an arbitrarily distributed sequence of replications and insertions
of random bits, provided the length of this sequence follows a sub-exponential
distribution.
