The existence conjecture for log canonical models

Let $\pi:(Y,B)\to S$ be a projective morphism. Assume that $(Y,B)$ has log canonical singularities, and that the restriction of $\cal O_Y(N(K+B))$ to each generic fiber is big, meaning that it contains an ample divisor.

Existence conjecture for log canonical models. The graded ring of $\cal O_S$-modules

$$\bigoplus_{d\ge 0}\pi_*\cal O_Y\bigl(dN(K_Y+B)\bigr)$$

is finitely generated.

Finite generation would provide the log canonical model needed in the properness argument for the moduli functor and moduli space. The source says that only this conjectural log minimal model input, in the one-dimensional semistable case, is needed there; it gives no resolution status.