 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