For a language $L$, define: $$ NE(L) = \{x \in L : x \text{ is not the proper prefix of any string in } L\} $$
I'm trying to show context-free languages are not closed under this operation. I've been struggling for a long time now trying to find a counterexample, that is, a language $L$ such that $L$ is context-free but $NE(L)$ is not context-free, and have come up with nothing. I'd appreciate ideas or hints about languages to look into.
Edit: For the vast majority of context-free languages, it seems that either $NE(L) = L$ or $NE(L) = \varnothing$. I'm having trouble even finding candidate languages.