Let $\mathcal{X}$ be a regular model of $X$ and $\mathcal{L}\in\Pic(\mathcal{X})$ a line bundle whose restriction to $X$ is ample. If $\mathfrak{a}$ is a vertical coherent ideal sheaf on $\mathcal{X}$ and $c>0$ is a rational number such that $\mathcal{L}\otimes\mathfrak{a}^{c}$ is nef, then we have

$$H^{q}\left(\mathcal{X},\omega_{\mathcal{X}}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}^{c})\right)=0\,\,\text{ for all }q\geq 1.$$

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

$$H^{q}\left(\mathcal{X},\omega_{\mathcal{X}}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}_{\bullet})\right)=0\,\,\text{ for all }q\geq 1.$$