Numerical abundance conjecture for klt pairs

Let $(X,\Delta)$ be a Kawamata log terminal (klt) pair. The numerical abundance conjecture asserts that the numerical and Iitaka dimensions of the log canonical divisor coincide:

$$\kappa_{\sigma}(X,\mathrm{K}_X + \Delta) = \kappa(X,\mathrm{K}_X + \Delta).$$

This conjecture is used to obtain an equivalence of Gelfand–Kirillov dimensions for the canonical ring of a maximal order and the log canonical ring of its centre. Its status is not resolved in the supplied source context.