The folklore local-to-global conjecture for cubic hypersurfaces

Let $X\subset \mathbf{P}_{\mathbf{Q}}^{n-1}$ be a cubic hypersurface with $n\geqslant 10$. The existence of points over every completion is necessary for a rational point, and in this range the real condition is automatic for a cubic form.

Folklore local-to-global conjecture. If $X(\mathbf{Q}_p)\neq\emptyset$ for every prime $p$, then

$$X(\mathbf{Q})\neq\emptyset.$$

Demyanov and Lewis proved the local conditions at every prime when $n\geqslant 10$, but the asserted global conclusion remains unresolved; the conjecture is the Hasse-principle expectation for cubic hypersurfaces in this range.