In this post, Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^*$, which is under the name: Kashiwara and Tanisaki - Characters of irreducible modules with non-critical highest weights over affine Lie algebras.
In other words, for complex semisimple Lie algebra, the problem for expressing $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M(\mu)$ is completely answered, where $M(\eta)$ is the Verma module with weight $\eta$ and $L(\eta)$ is its unique simple quotient.
What is about the problem for expressing $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M_I(\mu)$, where $M_I(\eta)$ is the parabolic Verma module with weight $\eta$ and $L(\eta)$ is its unique simple quotient. Is it still a Conjecture or is it just easily derived from the expression of $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M(\mu)$?
As pointed out by Rafael Mrđen, the above problem has been solved for arbitrary regular integral weights.
I would like to know the story for singular non-integral weights. Is it the same formula? I believe there should be some difference as in the case for ordinary Kazhdan-Lusztig Conjecture. For example, one need the integral Weyl group of $\lambda$: $W_{[\lambda]}:=\langle s_\alpha\in W:\alpha\in \Phi_{[\lambda]}\rangle$, where $\Phi_{[\lambda]}:=\{\alpha\in\Phi:\langle\lambda,\alpha^\lor\rangle\in\mathbb{Z}\}$ in order to state the character formula of $L(\lambda)$ for non-integral $\lambda$. I also believe that singularity may play a role in the formula.
I hope to see the explicit formula for $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M_I(\mu)$ and some kinds of parabolic KL polynomials, where $\lambda$ is singular non-integral. Any paper talks about this explicitly?
