Kawamata's effective non-vanishing conjecture for varieties with trivial first Chern class

Let $X$ be a smooth projective variety with $c_1(X)=0$ in $H^2(X,\mathbb{R})$, and let $L$ be a nef and big line bundle on $X$. Kawamata's trivial-first-Chern-class conjecture. Then

$$H^0(X,L)\neq 0.$$

This is the specialization of Kawamata's effective non-vanishing prediction to manifolds with trivial first Chern class. The paper proves it for several hyperkähler and Calabi--Yau cases, while it remains open in general.