A positive integral SL2 twist is regular but not D-affine

For $\alpha$ the unique positive root, $⟨\kappa ,{\alpha }^{\vee }⟩=c>0$, so $\kappa$ is regular and non-antidominant. By [F1], $M=\mathcal{O}(-c-1)$ is a left ${\mathcal{D}}_{\kappa }$-module, and it is nonzero because it is locally free of rank one.

With $z=y\mathrm{/}x$, $w=x\mathrm{/}y$ and transition $e_1=z^d{e}_0$ for $d=-c-1$, the Cech differential has image spanned by nonnegative powers $z^n$ and powers $z^{d-n}$ for $n\ge 0$. The first summand gives every exponent $n\ge 0$, the second every exponent $n\le -c-1$; the quotient has basis $z^{-c},z^{-c+1},\dots ,z^{-1}$, of size $c$. No exponent belongs to the intersection of those two ranges, so $H^0=0$. Thus $H^1(X,M)\cong {\mathbb{C}}^c$ and $\mathrm{\Gamma }(X,M)=0$.

Since $M$ is nonzero and quasi-coherent but $\mathrm{\Gamma }(M)=0$, the map $0\to M$ proves failure of conservativity. For the standard chart inclusions $j_0,j_1,j_{01}$, put $E=j_{0\ast }(M{\mathrm{\mid }}_{U_0})\oplus j_{1\ast }(M{\mathrm{\mid }}_{U_1})$ and $F=j_{01\ast }(M{\mathrm{\mid }}_{U_0\cap U_1})$. Restriction and difference give a ${\mathcal{D}}_{\kappa }$-linear exact sequence $0\to M\to E\to F\to 0$: differential operators act on direct images by restriction, the kernel consists of sections that glue, and on $U_i$ the other direct-image summand maps isomorphically onto $F{\mathrm{\mid }}_{U_i}$. These sheaves are quasi-coherent by [F4]: on $U_0=\mathrm{Spec}\mathbb{C}[z]$ their distinguished-open sections are the localizations of $\mathbb{C}[z]e_0$, $(\mathbb{C}[z]\oplus \mathbb{C}[z,z^{-1}])e_0$ and $\mathbb{C}[z,z^{-1}]e_0$, respectively; the same calculation holds with $w,e_1$ on $U_1$. The map $\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)$ is exactly the computed Cech differential with nonzero cokernel. Thus global sections is not exact on quasi-coherent ${\mathcal{D}}_{\kappa }$-modules.