The uniform interpolated lc rational polytope conjecture

Let $d$ be a positive integer, let $\Gamma_0\subset[0,1]\cap\mathbb{Q}$ be finite, and let $t_0\in[0,1]$ be irrational. Uniform interpolated lc rational polytope conjecture. There is a positive real number $\delta$, depending only on $d$, $\Gamma_0$, and $t_0$, such that whenever $(X,\mathcal{F},B^{\operatorname{ninv}}+(1-t_0)B^{\operatorname{inv}},t_0)$ is an lc adjoint foliated structure of dimension $d$ with $B\in\Gamma_0$, then

$$(X,\mathcal{F},B^{\operatorname{ninv}}+(1-t)B^{\operatorname{inv}},t)$$

is lc for every $t\in(t_0-\delta,t_0+\delta)$. This is proposed as a foliated analogue of uniform lc rational polytopes; related results are known in dimensions at most three and for algebraically integrable foliations.