Not true. It refers to how much vitality they have as a maximum, not currently. Anarion, Lord of Anorien (P) (0P58) is another example. With Narsil and the Saga he can make minions -5 until he is exhausted because he is a 5 vitality companion. It doesn't work like A Promise...
"Each wound token on a character depletes that character's vitality by 1."- official rules...
The maximum vitality shouldn't matter in this case. The total vitality possessed is depleted by the exertions/wounds, and that is what is counted, then, as the character's "vitality".
However, I'm not sure if, in this situation, the exertions are counted before the strength bonus is given.