= Solution
The centre commutes with $\mathfrak h$, so every nonzero <representation weight> component of the common <eigenvector> $m$ has the same <central character> $\theta$. Pick such a component $m_\lambda$. In <Category O>, its $U(\mathfrak n_+)$-span is finite-dimensional and has finitely many weights. Choose a maximal <representation weight> in that span and a nonzero vector $v$ of that <representation weight>, say $\nu$. Every <positive root> operator kills $v$, since it would raise the <representation weight> within the same finite span. Centrality ensures $zv=\theta(z)v$.
The universal property supplies a nonzero map $M^{\nu+\rho}\to M$ taking its highest vector to $v$. The centre acts on that image both by $\theta$ and by $\theta_{\nu+\rho}$. Hence
$$
\boxed{\theta=\theta_\xi,\qquad \xi=\nu+\rho,\quad\nu\in\operatorname{wt}(M).}
$$
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