show $H^q(\mathbb{Q}^n,\mathcal{S}^{\vee}\otimes\mathcal{E}(d+j))=0$ for all $q>0$ and $j\geqq -\lfloor\frac{n}{2}\rfloor+1$ and a spinor bundle $\mathcal{S}$ on $\mathbb{Q}^n$, it is enough to show that $H^q(\mathbb{Q}^n,\mathcal{S}\otimes\mathcal{E}(d+j))=0$ for all $q>0$ and $j\geqq -\lfloor\frac{n}{2}\rfloor$ and a spinor bundle $\mathcal{S}$ on $\mathbb{Q}^n$. We have an isomorphism $H^q(\mathbb{Q}^n,\mathcal{S}\otimes\mathcal{E}(d+j))\cong 
H^q(\mathbb{P}(\mathcal{S}), H(\mathcal{S})\otimes p^*(\mathcal{E}(d+j)))$, where $p:\mathbb{P}(\mathcal{S})\to \mathbb{Q}^n$ is the projection and $H(\mathcal{S})$ is the tautological line bundle on $\mathbb{P}(\mathcal{S})$. Let $\tilde{\pi}:\mathbb{P}(p^*\mathcal{E})\to \mathbb{P}(\mathcal{S})$ be the projection. Since $H(\mathcal{S})\otimes p^*(\mathcal{E}(d+j))\cong 
p^*\mathcal{E}\otimes H(\mathcal{S})\otimes p^*\mathcal{O}_{\mathbb{Q}^n}(d+j)$, we have an isomorphism $$H^q(\mathbb{P}(\mathcal{S}), H(\mathcal{S})\otimes p^*(\mathcal{E}(d+j)))
\cong
H^q(\mathbb{P}(p^*\mathcal{E}), H(p^*\mathcal{E})
+
\tilde{\pi}^*(H(\mathcal{S})
+
p^*\mathcal{O}_{\mathbb{Q}^n}(d+j))).$$ We claim here that the last cohomology group vanishes by the Kodaira vanishing theorem; first observe that $H(p^*\mathcal{E})+
\tilde{\pi}^*(H(\mathcal{S})+ p^*\mathcal{O}_{\mathbb{Q}^n}(d+j))-K_{\mathbb{P}(p^*\mathcal{E})}
$ is isomorphic to $$(r+1)H(p^*\mathcal{E})+
\tilde{\pi}^*(H(\mathcal{S})+ p^*\mathcal{O}_{\mathbb{Q}^n}(j)-K_{\mathbb{P}(\mathcal{S})}).$$ To show the last line bundle is ample, it is enough by the Nakai-Moishezon criterion to show that $H(\mathcal{S})+ p^*\mathcal{O}_{\mathbb{Q}^n}(j)-K_{\mathbb{P}(\mathcal{S})}$ is ample. To see this, recall that $\mathbb{P}(\mathcal{S})$ is a flag manifold parameterizing flags of one-dimensional and maximal dimensional linear subspaces of $\mathbb{Q}^n$; set $s=\lfloor \frac{n}{2}\rfloor$, and let $q:\mathbb{P}(\mathcal{S})\to S$ be the projection, which is a $\mathbb{P}^{s}$-bundle, to the spinor variety $S$. Recall also that $H(\mathcal{S})\cong q^*\mathcal{O}_S(1)$ for the ample generator $\mathcal{O}_S(1)$ of ${\mathop{\rm Pic}\nolimits}S$ (see, e.g., [@MR3275418 §5]). We see that $\mathbb{P}(\mathcal{S})$ is a Fano manifold of Picard 