Sadly, there are always different grades of immortality.
For example... I forgot which of the Norse gods it was, but nothing could kill him except holly. And Loki slays him with a sprig of holly.
So you can't just say 'Immortals will die with no head!'
It could be 'Immortals will be inconvenienced with no head!'
Or 'Immortals... we can't even cut off their head.'
Holly? Really? That's pretty weak.