Folk conjecture on sections of negative-square line bundles on K3^[2]-type varieties

Let $X$ be a hyperkähler manifold deformation equivalent to $K3^{[2]}$, and let $L$ be a line bundle on $X$ such that $q_X(L)=-2$.

Folk conjecture. If $(c_1(L),H^2(X;\mathbb Z))_X=\mathbb Z$, then either $L$ or $L^{-1}$ has a nonzero section. If $(c_1(L),H^2(X;\mathbb Z))_X=2\mathbb Z$, then either $L^2$ or $L^{-2}$ has a nonzero section.

The source labels this claim “Folk?” and offers no attribution or resolution status. It is connected with the expected role of negative-square classes in describing the ample cone of varieties of $K3^{[2]}$-type.