rt.representation theory - Are there elements of fixed weight in a crystal not killed by too many Kashiwara operators?

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

1. a weight $lambda$ of a simple Lie algebra $mathfrak g$, and


2. 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 $tilde{F}_alpha^{n_alpha}xneq 0$?

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