  We explain how the (shifted) Ratios Conjecture for $L(s,\chi)$ would extend a
randomization argument of Harper from a conductor-limited range to an unlimited
range of ``beyond square-root cancellation'' for character twists of the
Liouville function. As a corollary, the Liouville function would have
nontrivial cancellation in arithmetic progressions of modulus just exceeding
the well-known square-root barrier. Morally, the paper passes from random
matrices to random multiplicative functions.
