Voisin's ambientness conjecture for the triple-cycle Gamma

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a general hypersurface of degree $n+2$, and let $\Gamma\in A^{2n}(X\times X\times X)$ be the cycle defined in the source. Voisin's conjecture.

$$\Gamma\in\operatorname{Im}\bigl(A^\ast(\mathbb{P}^{n+1}\times\mathbb{P}^{n+1}\times\mathbb{P}^{n+1})\to A^\ast(X\times X\times X)\bigr).$$

The conjecture is used as a conditional input for proving an MCK decomposition for Calabi–Yau hypersurfaces; the source provides no resolution.