I'm having difficulty proving undecidability of "is this CFG prefix-free?". (this proof is given as problem 5.32b in Sipser 3rd edition).
Another thread has the very different question "is this regular language prefix-free". One of the answers in that thread claims to prove undecidability for CFLs (even though that did not explicitly ask for a proof of undecidability for CFLs) using the usual Post Correspondence Problem (PCP) method.
The proof in the other thread claimed (as of Oct 11, 2020) that a certain CFL with an endmarker $\#$ is prefix free if and only if there's no PCP solution. However, it seems to be possible to have prefixes in the CFL even without any PCP solutions, which would mean the proof is wrong.
I also tried using the fact that "non-emptiness of intersection of two CFGs" is undecidable. However, we are only able to use undecidability of CFGs being prefix-free to prove undecidability of intersection non-emptiness, not the other way around.