The lc-conjecture for very basic slc-trivial fibrations

Let $f\colon (X, B_X) \to (Y,D)$ be a very basic slc-trivial fibration with structure decomposition

$$D=K_Y+B_Y+M_Y.$$

Here $B_Y$ is the discriminant part and $M_Y$ is the moduli part. The lc-conjecture. There exists an effective $\mathbb Q$-divisor $\Delta\sim_{\mathbb Q}B_Y+M_Y$ on $Y$ such that $(Y,\Delta)$ is log canonical. The conjecture is known when $M_Y$ is $\mathbb Q$-Cartier and numerically trivial, while the general case remains open.