Irreducibility of quadratic hulls of osculating varieties of closed orbits

Let $G$ be a connected complex semisimple algebraic group, let $V_{\lambda }$ be an irreducible $G$-module of highest weight $\lambda$, and let $$X_{d\lambda }=G\cdot [v_{d\lambda }]\subset \mathbb{P}{V}_{d\lambda }$$ be the closed orbit of a highest weight vector. We study the irreducibility of the zero locus of the quadratic equations containing the $p$-osculating variety $T^p{X}_{d\lambda }$. We prove that, if $d>4p$, its quadratic hull coincides with the osculating variety itself: $$\mathcal{Q}(T^p{X}_{d\lambda })=T^p{X}_{d\lambda }\mathrm{.}$$ In particular, the quadratic hull is irreducible. Thus, for every fixed $p$, sufficiently high Cartan powers of a closed highest weight orbit have $p$-osculating varieties determined set-theoretically by their quadratic equations.