$D$-affinity of Quadrics Revisited

Let $K$ be aa algebraically closed field of characteristic $p\geq3$ and let $Q_{n}\subset\mathbb{P}^{n+1}_{K}$ be a smooth quadric hypersurface. We show that if $n=2m\geq4$ then $Q_{n}$ is not $D$-affine. In particular, we show the grassmannian ${Gr}(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\geq3$. Our result complements previous work of A. Langer, who showed that if $p\geq n=2m+1$ then $Q_{n}$ is $D$-affine.