Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and $M(\lambda)$ be the corresponding Verma module. Then $M(\lambda)$ admits a Jantzen filtration, as explained e.g. in Section 3.1 of Kubel's thesis.

When $\mathfrak{g}$ is of finite type, $\lambda = w \cdot \nu := w(\nu + \rho) - \rho$, where $\nu$ is a dominant integral weight and $\rho$ is half the sum of the positive roots, it is known that the subquotients of the Jantzen filtration are semisimple, and their isotypic multiplicities are given by coefficients of Kazhdan-Lusztig polynomials (see e.g. Chapter 8 of the book of Humphreys on Category $\mathcal{O}$).

My question is: what is known/expected about the semisimplicity and isotypic multiplicities of subquotients of the Jantzen filtration in infinite type? I would also be interested in what is known/expected in finite type for more general $\lambda$.

As motivation/a related question, a very special case of the finite type story I would be interested in knowing in general is the following: let $s_i$ be a simple reflection, and suppose $s_i \cdot \lambda \leqslant \lambda$, i.e. $(\check{\alpha}_i, \lambda) \in \mathbb{Z}_{\geqslant 0}$, where $\check{\alpha}_i$ denotes the corresponding simple coroot. Let $L(\lambda)$ be the simple module of highest weight $\lambda$, and consider the short exact sequence: $$0 \rightarrow N(\lambda) \rightarrow M(\lambda) \rightarrow L(\lambda) \rightarrow 0.$$I want that $\text{Hom}(N(\lambda), L(s_i \cdot \lambda)) \neq 0$ -is this true? Note that in finite type, for $\lambda = w \cdot \nu$ as above, this follows from the semisimplicity and isotypic multiplicities of the second subquotient of the Janzten filtration (where I call $L(\lambda)$ the first subquotient).

Thank you for your help, and please let me know if anything is unclear!