Let $\mathcal{B}$ be a given very ample line bundle such that $\mathcal{A}:=\omega_{\mathcal{X}}\otimes\mathcal{B}^{n+1}$ is ample. By the Castelnuovo-Mumford criterion it is enough to check that

$$H^{q}\left(\mathcal{X},\mathcal{A}\otimes\mathcal{L}\otimes\mathcal{B}^{-q}\otimes\mathcal{J}(\mathfrak{a}^{c})\right)=0$$

for $q=1,\dots,n$, and this is a consequence of Theorem B.5. ∎