The strong Lefschetz conjecture for Milnor algebras

Let $F\in S_d$ be a smooth homogeneous polynomial, let $M(F)$ be its Milnor algebra, and set $T=(n+1)(d-2)$. For an integer $k\in[0,T/2)$ and a generic linear form $\ell\in S_1$, consider the multiplication map

$$\ell^{T-2k}: M(F)_k\to M(F)_{T-k}.$$

Strong Lefschetz conjecture. For any smooth $F\in S_d$, any integer $k\in[0,T/2)$, and any generic linear form $\ell\in S_1$, the induced multiplication map is an isomorphism. The strong Lefschetz property is a standard expected property of Milnor algebras; the source describes this as a well-known conjecture and refers to a brief overview, with no resolution supplied here.