Let $X \subset \mathbb P(\mathfrak g)$ be an adjoint variety for a simple complex Lie algebra $\mathfrak g$ appearing in the last line of the Freudenthal Magic Square, that is $\mathfrak g \in \{F_4,E_6,E_7,E_8\}$. In particular $\dim X=6m+9$ where $m \in \{1,2,4,8\}$. Consider minimal embedding of the VMRT of $X$ at a point $x$, that is $$ Y^{3m+3} \subset \mathbb P^{6m+7} \subset \mathbb P^{6m+8}=\mathbb P(T_{X,x}). $$ On the other hand we have the contact structure $0 \to F \to T_X \to L \to 0$, and the restriction of such a sequence to $x$ allow us to write $Y \subset \mathbb P(F_x)$.
Now, $F_x$ is endowed with a symplectic form $\omega$ that is the restriction to $F_x$ of the map $\omega=d\theta: F \times F \to L$. In particular $\mathbb P(F_x)$ has a symplectic structure. We blow-up $\mathbb P(F_x)$ along $Y$, obtaining a map $$ b: Bl_Y \mathbb P(F_x) \longrightarrow \mathbb P(F_x). $$ Question: do the map $b$ preserve the symplectic structure? More precisely, can we define a symplectic structure on $Bl_Y \mathbb P(F_x)$?
Any hint of a possible argument for the proof?
