BACKGROUND
Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X \to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity on points and whose comorphism is the $p^{th}$ power map. We say that $X$ is Frobenius split if the $p^{th}$ power morphism $ \mathcal O_X \to F_* \mathcal O_X $ splits (in the category of sheaves of $\mathcal O_X$-modules).
Let $G$ be a semisimple algebraic group over $k$ and let $B \subseteq G$ be a Borel subgroup. A number of varieties associated to $G$ are known to be Frobenius split: the flag variety $G/B$, the cotangent bundle $T^*(G/B)$, etc.
QUESTION
Is it known in general whether or not the tangent bundle $T(G/B)$ to $G/B$ is Frobenius split? I think that the splitting of $T(G/B)$ would be implied by a splitting of $X \times X$ that maximally compatibly splits the diagonal, so by recent work of Lauritzen and Thomsen (in fact, the proof in type $C$ was just posted on the arXiv), I believe one does know that $T(G/B)$ is Frobenius split in types $A$ and $C$. Does anyone know of any other work exploring this topic?
