Definable-envelope conjecture for nilpotent and soluble subgroups of NTP2 groups

Let $G$ be an $\operatorname{NTP}_{2}$ group and let $H$ be a subgroup of $G$.

Definable-envelope conjecture. If $H$ is nilpotent (respectively, soluble), then there is a definable nilpotent (respectively, soluble) group finitely many of whose translates cover $H$. If $H$ is additionally normal, then there is a definable normal nilpotent (respectively, soluble) group containing $H$.

This conjecture seeks an $\operatorname{NTP}_{2}$ analogue of definable-envelope results known for groups in $\operatorname{NIP}$ and simple theories. The source presents it as a natural direction for further research, without resolving it.