Symplecticity conjecture for parabolic partial implosions

Let $G$ be a connected semisimple group, let $P$ be a parabolic subgroup of $G$, and let $U_P$ be its unipotent radical. Let $B$ be a Borel subgroup and set

$$E=G\times_B\mathfrak{u}_P^{\perp}$$

as a homogeneous vector bundle over $G/B$. The projectivization $\mathbb{P}E^*$ is a projective bundle. Parabolic partial-implosion conjecture. The projective bundle $\mathbb{P}E^*$ is of Fano type. By the preceding theorem, this is equivalent to the symplecticity of $\overline{T^*(G/U_P)}$. The conjecture is supported by the evidence in the cited proposition; it has been confirmed for $G=\operatorname{SL}_n$ and $G=\operatorname{GL}_n$.