The relevant paper is "Local rigidity of quasi-regular varieties" by Pasquier and Perrin,
Math. Z. 265 (2010), no. 3, 589–600. They construct a smooth fibration over $\mathbb{C}$ such that the fiber over $t\neq 0$ is a orthogonal grassmannian $\mathbb{G}_q(2,7)$, but the fiber over 0 is not homogeneous.