Injectivity conjecture for primitive quantum algebras of Fano hypersurfaces

Let $j:Y\hookrightarrow X$ be the natural inclusion of a smooth Fano hypersurface $Y$ in a Fano manifold $X$ of Picard number $1$, with $\dim Y>2$. Let $r_X$ and $r_Y$ be the respective indices, let $h_X$ and $h_Y$ denote the hyperplane classes, and let $\mathcal{A}_{\perp}^0(X)$ and $\mathcal{A}_{\perp}^0(Y)$ be the corresponding primitive algebras. Write $\widehat{h_X}^{r_X}$ and $\widehat{h_Y}^{r_Y}$ for the associated operators.

Injectivity conjecture. With the same notation as in the quantum Lefschetz proposition, there is a natural injective morphism of algebras

$$\mathcal{A}_{\perp}^0(X)\longrightarrow \mathcal{A}_{\perp}^0(Y)$$

that intertwines the operators $\widehat{h_X}^{r_X}$ and $\widehat{h_Y}^{r_Y}$.

This predicts that the primitive quantum algebra of a smooth Fano hypersurface contains the corresponding primitive algebra of its ambient manifold in a way compatible with the index-power hyperplane operators. The supplied text gives no resolution status for this assertion.