Characteristic-p Lefschetz conjecture for codimension-three Artinian Gorenstein algebras

Let $A$ be an Artinian Gorenstein algebra of codimension three, with socle degree $j_A$, over an infinite field $\mathsf{k}$ of characteristic $p>j_A$. For a general enough linear form $\ell\in A_1$, consider the multiplication map $\ell^{j_A-2}:A_1\to A_{j_A-1}$. Characteristic-p Lefschetz conjecture. The map

$$\ell^{j_A-2}:A_1\to A_{j_A-1}$$

is an isomorphism. The conjecture isolates the characteristic-zero input in the paper's argument and predicts the remaining Lefschetz assertion in characteristic larger than the socle degree; its resolution is not supplied here.