Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group.

My question concerns specialisations of flag varieties over discrete valuation rings. Namely, let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$. Assume for simplicity that $K$ and $k$ are algebraically closed.

Let $X \to \textrm{Spec} R$ be a smooth projective scheme over $R$, and assume that the generic fibre is a flag variety. Is the special fibre also a flag variety?

Some remarks:

- If the generic fibre is isomorphic to a projective space, then the answer to my question is yes. This is Theorem 2 of [1].
- The converse to my question has a positive answer. Namely, if the special fibre is a flag variety, then the generic fibre is a flag variety. This is Proposition 5 of [2].

[1] Goren - Characterization and algebraic deformations of projective space.

[2] Démazure - Automorphismes et déformations des variétés de Borel.