On log canonical divisors that are log quasi-numerically positive

Let $(X,\mathrm{\Delta })$ be a four-dimensional log variety that is projective over the field of complex numbers. Assume that $(X,\mathrm{\Delta })$ is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisor $K_X+\mathrm{\Delta }$ is log quasi-numerically positive on $(X,\mathrm{\Delta })$ then it is semi-ample.