Base-point-freeness conjecture for lc algebraically integrable foliations

Let $(X,\mathcal{F},B)/U$ be an lc algebraically integrable foliated triple. Let $A$ be an ample$/U$ $\mathbb{R}$-divisor on $X$ such that $K_{\mathcal{F}}+B+A$ is nef$/U$. Then:

1. $K_{\mathcal{F}}+B+A$ is semi-ample$/U$.
2. If $K_{\mathcal{F}}+B+A$ is Cartier, then $\mathcal{O}_X(m(K_{\mathcal{F}}+B+A))$ is globally generated$/U$ for any integer $m\gg 0$.

Base-point-freeness conjecture. Under these hypotheses, the divisor $K_{\mathcal{F}}+B+A$ is semi-ample$/U$, and in the Cartier case its sufficiently high multiples are globally generated$/U$.

This is the foliated analogue of a base-point-free theorem for lc algebraically integrable foliations. The surrounding results provide positive evidence in locally stable-family and co-rank-one situations, while the general assertion remains open.