Properties of a highest weight module.

Question

Let $V$ be a highest weight module of a semisimple Lie algebra $\mathfrak{g}$ with highest weight $\lambda$. Then there is a highest weight vector $v$ such that $V=U(\mathfrak{g})^{-}\cdot v$. Let $x_i^{-}$ ($i\in I ={1, \ldots, n}$) be a set of generators of $U(\mathfrak{g})^{-}$. Let $W=\oplus_{\mu\in \mathfrak{h},v(\mu)\leq p}V_{\mu}$, where $\mathfrak{h}$ is generated by coroots $\alpha_1^{\vee},\ldots, \alpha_n^{\vee}$, and $v(\mu)=b_1+\ldots +b_n$ for $\mu=\sum_{i=1}^{n}b_i \alpha_i^{\vee}$. Does $W$ has the property: $\oplus_{\mu\in\mathfrak{h},v(\mu)\leq p+1}V_{\mu}\subset \sum_{i\in I} x_i^{-}\cdot W$? If $V$ is not a highest weight module, does $W$ have this property? Thank you very much.