Antidominant twists are D-affine and regular ones are conservative

Let $i>0$ and write as the filtered union of coherent ${\mathcal{O}}_X$-submodules from [F1]. Fix one ${\mathcal{F}}_j$. Choose a sufficiently dominant integral $\eta$ so that $H^i(X,{\mathcal{F}}_j\otimes L)=0$ by [F2], with $L={\mathcal{O}}_X(-\eta )$. The composite $H^i(X,{\mathcal{F}}_j)\to H^i(X,M)\to H^i(X,M\otimes L\otimes V)$ factors through $H^i(X,{\mathcal{F}}_j\otimes L\otimes V)$ by naturality of the coevaluation map. This last group is $H^i(X,{\mathcal{F}}_j\otimes L)\otimes V=0$ because $V$ is finite-dimensional. By [F4], the map $H^i(X,M)\to H^i(X,M\otimes L\otimes V)$ is injective. Thus $H^i(X,{\mathcal{F}}_j)\to H^i(X,M)$ is zero for every $j$. Taking the filtered colimit and using [F1] gives $H^i(X,M)=0$.

For every quasi-coherent left ${\mathcal{D}}_{\kappa }$-module $M$, [F5] and step 1.1 give $R^q{\mathrm{\Gamma }}_{\mathrm{Mod}({\mathcal{D}}_{\kappa })}(X,M)\cong H^q(X,M)=0$ for each $q>0$. Given a short exact sequence $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of quasi-coherent ${\mathcal{D}}_{\kappa }$-modules, view it in the full module-sheaf category. Its derived-functor long exact sequence and the vanishing of $R^1\mathrm{\Gamma }(M^{\mathrm{\prime }})$ show that $\mathrm{\Gamma }(X,M)\to \mathrm{\Gamma }(X,M^{\mathrm{\prime }\mathrm{\prime }})$ is surjective; left exactness supplies exactness at the other terms. Hence $\mathrm{\Gamma }$ is exact on quasi-coherent ${\mathcal{D}}_{\kappa }$-modules for antidominant $\kappa$.

Now assume $\kappa$ is regular antidominant and $\mathrm{\Gamma }(X,M)=0$. For any coherent ${\mathcal{F}}_j\subseteq M$, choose a sufficiently dominant integral $\eta$ so that ${\mathcal{F}}_j\otimes L$ is globally generated by [F3]. By the split surjection in [F4], $\mathrm{\Gamma }(X,M\otimes L)$ is a direct summand of $\mathrm{\Gamma }(X,M\otimes V^{\ast })=\mathrm{\Gamma }(X,M)\otimes V^{\ast }=0$; hence $\mathrm{\Gamma }(X,M\otimes L)=0$. The inclusion ${\mathcal{F}}_j\otimes L\hookrightarrow M\otimes L$ remains injective on global sections because $\mathrm{\Gamma }$ is left exact, so $\mathrm{\Gamma }(X,{\mathcal{F}}_j\otimes L)=0$. A globally generated sheaf with no global sections is zero, hence ${\mathcal{F}}_j=0$. All coherent subsheaves in [F1] vanish, so $M=0$. This applies the splitting to $M$, not to the arbitrary coherent ${\mathcal{O}}_X$-submodule.