NIP trace-interpretation conjecture for infinite groups

A structure $\mathscr{M}$ is $\aleph_0$-categorical if its complete theory has a unique countable model up to isomorphism, and it is NIP if its theory has no independence property. An infinite group may be interpreted or trace defined in $\mathscr{M}$ using the paper's respective notions.

NIP group conjecture. If $\mathscr{M}$ is $\aleph_0$-categorical and NIP, then $\mathscr{M}$ interprets an infinite group if and only if $\mathscr{M}$ trace defines an infinite group.

The surrounding discussion contrasts ordinary interpretation with trace definability and gives this equivalence as an open conjectural principle.