Arithmetic unlikely intersections in powers of the multiplicative group

Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its size" should not be too big compared to the complexity" of the subgroup scheme. We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-Mérai-Ostafe-Sha. We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.