The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple
$\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$. 

The Verma module is of the
form $M(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$, where $\mathbb{C}_\lambda$ is a simple $\mathfrak{b}$-module with weight $\lambda$. Denote by $L(\lambda)$ the unique
simple quotient of $M(\lambda)$.

The parabolic Verma module is defined to be $M_I(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{p})} F(\lambda)$,
where $F(\lambda)$ is the simple finite-dimensional $\mathfrak{l}$-module with highest weight $\lambda$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is defined to be $\Lambda^+_I = \{\lambda \in \mathfrak{h}^*  :  \langle\lambda,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}$.


By Proposition 9.3 and Theorem 9.4 in ***Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$***, we get $L(\lambda)\in\mathcal{O}^\mathfrak{p}\iff \lambda\in\Lambda_I^+$.
Also $\lambda\in \Lambda_I^+ \implies M_I(\lambda)\in \mathcal{O}^\mathfrak{p}$.

My question: What is about $M(\lambda)$? 
Does $\lambda\in \Lambda_I^+$ implies $M(\lambda)\in \mathcal{O}^\mathfrak{p}$? If not, any counterexample?