The sheaf Lefschetz injectivity conjecture for irreducible holomorphic symplectic fourfolds

Let $X$ be an irreducible holomorphic symplectic $4$-fold, let $E$ be a nef line bundle on $X$, and let $L^*$ denote the map in the sheaf Lefschetz action. The sheaf Lefschetz injectivity conjecture. The map

$$L^*:\mathbb H^2(X,E)\longrightarrow\mathbb H^0(X,E)$$

is injective. The conjecture is suggested by calculations for examples: the analogous map fails to be injective in dimension six, while a more detailed analysis suggests injectivity in dimension four; no proof is supplied in the source.