It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ has a direct sum decomposition $M=\oplus M_i$ such that all weights of each $M_i$ are contained in a single coset of root lattice $\Lambda_r$ in $\mathfrak{h}^*$. Then, the category $\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ decomposes as a direct sum of full subcategories, which can be indexed by the nonempty intersection of the orbit $W\cdot\lambda$ with the cosets $\mathfrak{h}^*/\Lambda_r$. From now on, we use the anti-dominant weight $\mu$ in the intersection to parameterize the corresponding subcategory of $\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$. We denote this subcategories by $\mathcal{O}^\mathfrak{p}_{\mu}$. In particular, if $I=\emptyset$, then the parabolic subalgebra $\mathfrak{p}$ collapses to a Borel subalgebra $\mathfrak{b}$ of $\mathfrak{g}$ and $\mathcal{O}^\mathfrak{p}_{\mu}$ turns into the ordinary block $\mathcal{O}_{\mu}$ of the BGG category $\mathcal{O}$.

I would like to know whether $\mathcal{O}^\mathfrak{p}_{\mu}=\mathcal{O}^\mathfrak{p}_{\chi_\mu}$ for $\mu$ being regular integral?