Ionescu's effective non-vanishing conjecture for quasi-polarized manifolds

Let $X$ be a smooth projective variety of dimension $n$ over the complex numbers, and let $L$ be a nef and big line bundle on $X$. Thus $(X,L)$ is a quasi-polarized manifold. Ionescu's effective non-vanishing conjecture. If $K_X+L$ is nef, then

$$h^0(K_X+L)>0.$$

This conjecture asks for an effective non-vanishing result for the adjoint bundle of a quasi-polarized manifold. It is stated as an open problem in the source.