Stable irrationality conjecture for very general cubic hypersurfaces

Let $n\geq 3$ and let $X\subset \mathbb P^{n+1}$ be a very general cubic hypersurface. Stable irrationality conjecture. $X$ is not stably rational. The conjecture asks whether the known stable irrationality bounds for very general hypersurfaces can be made constant in the cubic case; it is open even in low dimensions, including the case of cubic threefolds.