Uniform rational polytopes conjecture for functional divisors on foliations

Let $v_1^0,\dots,v_m^0$ be positive integers and set $\boldsymbol{v}_0=(v_1^0,\dots,v_m^0)$. The rational envelope of $\boldsymbol{v}_0$ is the ambient rational-affine object in which an open neighborhood of $\boldsymbol{v}_0$ is taken. A uniform rational polytopes conjecture for functional divisors on foliations asserts that there exists an open set $U\ni\boldsymbol{v}_0$ in the rational envelope of $\boldsymbol{v}_0$ such that, for any lc foliated triple of dimension at most $3$ of the form

$$(X,\mathcal{F},B(\boldsymbol{v}_0))\quad\text{with}\quad B(\boldsymbol{v}_0):=\sum_{i=1}^m v_i^0B_i,$$

where $B_i\geq 0$ are distinct Weil divisors, the triple

$$(X,\mathcal{F},B(\boldsymbol{v}))\quad\text{with}\quad B(\boldsymbol{v}):=\sum_{i=1}^m v_iB_i$$

is lc for every $\boldsymbol{v}=(v_1,\dots,v_m)\in U$.

This is proposed as the uniform rational-polytope input needed to prove the global ACC conjecture for foliated threefolds. It concerns stability of log canonicity under nearby coefficient vectors and remains open in the foliated setting.