Let $T \subset B \subset G$ be a reductive group, a Borel subgroup and a maximal torus over a field of characteristic $0$, with respective Lie algebras $\mathfrak{b} \subset \mathfrak{g}$. For a weight $\lambda \in X^{*}(T)$ let $M(\lambda):=U\mathfrak{g} \otimes_{U \mathfrak{b}} \lambda$ be the Verma module of weight $\lambda$, it has a filtration $M^{\le n}(\lambda)$ given by the filtration on $U \mathfrak{g}$, and each filtered piece is a $B$-representation. My question is: is it always the case that $M^{\le n}(\lambda)$ is a subquotient (as a $B$-represenation) of the restriction of a $G$-representation?
I believe one can take the $G$-representation to be a highest weight module (a Weyl module) for a "sufficiently dominant" (for $G=\text{GL}_2$ it corresponds to taking $\text{Sym}^n \mathbb{C}^2$ for $n$ big enough) $\mu$ with respect to $\lambda$, and take the $B$-submodule to be the one generated by the vectors whose weight correspond to the vertices that are minimal (in the usual ordering of weights given by the choice of Borel) in the convex hull of weights that is $M^{\le n}(\lambda)$. I'm pretty sure this works for $\text{GL}_2$, but I'm not confident enough to make this sketch into an actual proof. I would appreciate a reference for a result with a similar technique.
