The non-vanishing conjecture for canonical sheaves

Let $X$ be a smooth projective variety. A $\mathbb{Q}$-divisor is effective if it is a nonnegative $\mathbb{Q}$-linear combination of prime divisors, and two $\mathbb{Q}$-divisors are $\mathbb{Q}$-linearly equivalent when their difference is the divisor of a rational function after multiplication by a positive integer. Non-vanishing conjecture. The canonical sheaf $\omega_X$ is pseudo-effective if and only if it is $\mathbb{Q}$-linearly equivalent to an effective $\mathbb{Q}$-divisor. This conjecture concerns the existence of effective representatives of the canonical class and is a basic problem in the minimal model program; the supplied source does not state a resolution, so its status is recorded as open.