I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely solved in characteristic $p$, and using Scholze's perfectoid spaces, also for hypersurfaces of toric varieties.

What is it about pure $\mathbb{Q}_p$-motives that makes the conjecture that much harder?