Uniform boundedness of decimal multiplicative persistence

Prove that every positive decimal integer has multiplicative persistence at most 11.

Background

Multiplicative persistence in base 10 is obtained by repeatedly replacing a positive integer with the product of its decimal digits until a single decimal digit is reached. The paper records the classical conjecture that this process never requires more than 11 iterations. The smallest currently known example with persistence 11 is 277777788888899, and no example with larger persistence is known.

The paper resolves the related 2-adic obstruction for nonzero even terminal digits, but it does not establish the global persistence bound asserted in Conjecture A. Thus, the conjecture remains an unresolved problem beyond the results proved in the paper.