The too long, didn't read form of the question would simply be, has someone completed A. Joseph's proof of the Demazure character formula? Is Jacobson's proof considered complete?

In more detail, Joseph's paper "On the Demazure character formula" only proves the formula for finite $\mathfrak{g}$-modules of the form $L(\lambda)$ where $\lambda$ is 'sufficiently large'. The Demazure operators make their appearance in the lemma 2.5, where for a finite $\mathfrak{b}$-module $F$, one has the character formula, $$ ch \mathscr{D}_\alpha F = \Delta_\alpha (ch(Im(F\to \mathscr{D}_\alpha F))). $$ So the difficulty in the proof of the full formula for $L(\lambda)$ is in showing that at each stage the natural $\mathfrak{b}$-module map, $$ \mathscr{D}_{\alpha_{i}}\cdots \mathscr{D}_{\alpha_{1}} (\mathbb{C}_\lambda) \to \mathscr{D}_{\alpha_{i+1}} \mathscr{D}_{\alpha_{i}}\cdots \mathscr{D}_{\alpha_{1}} (\mathbb{C}_\lambda) $$ is injective, where $w = s_{\alpha_n}\cdots s_{\alpha_1}$ is a reduced decomposition of the longest element of the Weyl group, and the $\mathscr{D}_{\alpha_{i}}$ are related the Zuckerman functors.

It isn't expressly mentioned in Kashiwara's paper, "The crystal base and Littelmann's refined Demazure character formula", but is it known if injectivity follows from Kashiwara's 'string property'? This seems immediate, but the review of Joseph's paper on mathscinet makes it sound as though Joseph's proof was never finished.