The log-canonical abundance conjecture

Let $(X\vert\Delta)$ be a log-canonical pair. Define the numerical dimension $\nu(X,D)$ for a rational divisor $D$ by asymptotic growth of sections after adding an ample divisor. Abundance conjecture. One has

$$\nu(X,K_X+\Delta)=\kappa(X,K_X+\Delta).$$

This is a numerical-dimension form of abundance, relating numerical positivity to the Kodaira dimension of the log canonical divisor. The source states it as a conjectural form and gives no resolution.