Let $\mathcal{X}$ be a regular model. Then there exists an ample line bundle $\mathcal{A}$ on $\mathcal{X}$ such that the following holds. Given $\mathcal{L}\in\Pic(\mathcal{X})$, a vertical ideal sheaf $\mathfrak{a}$ and a rational number $c>0$ such that $\mathcal{L}\otimes\mathfrak{a}^{c}$ is nef, the sheaf

$$\mathcal{A}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}^{c})$$

is globally generated. In particular, if $\mathfrak{a}_{\bullet}$ is a graded sequence of vertical coherent ideal sheaves on $\mathcal{X}$ such that $\mathcal{L}^{m}\otimes\mathfrak{a}_{m}$ is globally generated for all sufficiently divisible $m$, then

$$\mathcal{A}\otimes\mathcal{L}^{m}\otimes\mathcal{J}(\mathfrak{a}_{\bullet}^{m})$$

is globally generated for all $m$.