Curious. Do you know if the carrot got stuck
within the neck where it could been seen bulging within the neck or further down?
Then if it could be seen bulging, do you know of a way to dislodge it? Even if it wasn't visible, but one knew what was happening.
If not, maybe ask your vet next time you see him/her.
I know on the thread about the sad event has ideas of dislodging, but the vet might have a safer, faster, appropriate way.