The number system in rational base $3\mathrm{/}2$ and the $3x+1$ problem

The representation of numbers in rational base $p\mathrm{/}q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p\mathrm{/}q=3\mathrm{/}2$. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$ function. Our purpose in this note is to expose these links and motivate further research into them.