I've come across an annoying lemma trying to finish up an argument, and I was hoping one of you guys knew about it.

**Question:** Given

- a weight $\lambda$ of a simple Lie algebra $\mathfrak g$, and
- integers $n_\alpha$ for each simple root $\alpha$,

Is there a highest weight $\nu$, such that in the crystal of with highest weight $\nu$ there is an element $x$ of weight $\lambda$ such that $\tilde{F}_\alpha^{n_\alpha}x\neq 0$?

This is true in $\mathfrak{sl}_2$, which makes me hopeful about other Lie algebras, but the argument isn't coming together for me.