A Jew, a Christian and a Muslim walk into a restaurant....
... and find someone dying on the floor.
You see the problem straight away - if you have a one-on-one situation, then I can see that your question seems debatable. But there's more than one faith and many will insist they're exclusively the one to go for. Incidentally I'm not really changing the question here because in your case there are two people, potentially of their own differing faith. Who'se to say the dying person doesn't convert the well one's mind ?
Also the 'close to death' part is only just relevant, in that it lends gravity to the situation, but the same question applies to well people who aren't so close to death.
I'd say the ethics are in a sense subjective:
From each individual's point of view (the ones doing the persuading), they think they're doing the 'right thing' by persuading the dying person to go with their faith, for example to save them from hell (or equivalent). This assumes that these people put this notion of saving someone from the 'wrong faith' above the ethical ranking of allowing someone to be what they are.
I should make the point that this is about "having a faith A versus faith B", the point being a change from one to the other, not which way or what you end up with.
From the group point of view, or an independent (atheist?) observer, or the person on the floor, each argument to change to a given faith is cancelled by any argument to change to any other faith. I'd imagine this would make the circumstances almost laughably irrelevant.
So back to the persuading individual's point of view: Is this ethical ?
No, because objectively, there's no discernable value in one faith over another, and so there's no value in changing someone's mind. It would just amount to a person asserting their own belief on someone else.