Mukai's conjecture on syzygies of adjoint line bundles on surfaces

Let $X$ be a surface and let $A$ be an ample line bundle on $X$. For an integer $p\geq 0$ and an integer $\ell$, set $L=K_X+\ell A$. Mukai's conjecture. The line bundle $L$ satisfies Property-$(N_p)$ for all $\ell\geq p+4$. This is described as an open problem generalizing Green's theorem on syzygies of curves to surfaces; the asserted bound concerns adjoint linear series on arbitrary surfaces.