Existence of log terminal models for compact Kähler gklt pairs

Let $(X,B+\boldsymbol{\beta})$ be a compact Kähler gklt pair such that $K_X+B+\boldsymbol{\beta}_X$ is pseudo-effective.

Existence of log terminal models. The pair $(X,B+\boldsymbol{\beta})$ has a log terminal model.

This is proposed as an extension of the paper's main results to gklt pairs without assuming that $B+\boldsymbol{\beta}_X$ is modified big. The conjecture would follow from termination of flips in the Kähler setting.