  The goal of this work is to prove an analogue of a recent result of Harper on
almost sure lower bounds of random multiplicative functions, in a setting that
can be thought of as a simplified function field analogue. It answers a
question raised in work of Soundararajan and Zaman, who proved moment bounds
for the same quantity in analogy to those of Harper in the random
multiplicative setting. Having a simpler quantity allows us to make the proof
close to self-contained, and perhaps somewhat more accessible.
