Debarre–Manivel's conjecture for Fano varieties of lines on cubics

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth cubic, and let $F=F(X)$ be its Fano variety of lines. Let $G:=G(1,\mathbb{P}^{n+1})$ be the Grassmannian of lines. Assume $\ell\in\mathbb{N}$ satisfies

$$\frac{(\ell+4)(\ell+3)}{2}\le n+1.$$

Debarre–Manivel's conjecture. The inclusion $F\hookrightarrow G$ induces an isomorphism

$$A_\ell(F)\cong A_\ell(G),$$

and, in particular, $A_\ell^{hom}(F)=0$. This is a special case of a broader conjecture concerning Fano varieties of linear subspaces in complete intersections. The statement is presented as conjectural in the source and is open in this generality.