As is shown in ** Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$**, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each of which is a highest weight module in $\mathcal{O}^\mathfrak{p}$.
Thus the action of $Z(g)$ on $M$ is finite.

Let $ M^\chi = \{v \in M \ | \ (z - \chi(z))^nv = 0 \ \text{for some $n \in \mathbb{Z}_{>0}$ depending on $z$}\}, $ then $z - \chi(z)$ acts locally nilpotently on $M^\chi$ for all $z \in Z(\mathfrak{g})$ and $M^\chi$ is a $U(\mathfrak{g})$-submodule of $M$.

Denote by $\mathcal{O}^\mathfrak{p}_\chi$ the full subcategory of $\mathcal{O}^\mathfrak{p}$ whose objects are of the form $M^\chi$, then we have the following direct sum decomposition $ \mathcal{O}^\mathfrak{p}=\bigoplus_{\chi}\mathcal{O}^\mathfrak{p}_\chi, $ where $\chi = \chi_\lambda$ for some $\lambda \in\mathfrak{h}^*$.

Because $M$ is generated by finitely many weight vectors, it must therefore be the direct sum of finitely many nonzero submodules $M^\chi$.

We call $\lambda,\mu\in\mathfrak{h}^*$ are linked if $\lambda=w\cdot\mu:=w(\mu+\rho)-\rho$ for some $w\in W$ where $\rho$ is the half sum of positive roots and $W$ is the Weyl group of $\Phi$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is $ \Lambda^+_I = \{\lambda \in \mathfrak{h}^* : \langle \lambda, \alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}, $ there is a fact that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$ iff $\lambda\in\Lambda^+_I$.

Then here the question, let $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$, then denote $\Phi$ the root system and $\Delta=\{\alpha,\beta\}$ the set of simple roots in $\Phi$. Consider $t\alpha+(1-t)(\alpha+\beta)\in \Lambda_I^+$ with $0\le t\le 1$. Then there should have infinitely many non-linked weights (since $t$ infinite and $t\alpha+(1-t)(\alpha+\beta)$ are weights lie between $\alpha$ and $\alpha+\beta$) such that $L(t\alpha+(1-t)(\alpha+\beta))\in\mathcal{O}^\mathfrak{p}_{\chi_{t\alpha+(1-t)(\alpha+\beta)}}$,

which then should gives infinitely many $\mathcal{O}^\mathfrak{p}_{\chi}$.

Can anyone give me some examples/ explain why my intuition fails to convince me that there is indeed finitely many $\Phi^+_I$-dominant integral weights up to linkage?