Just noticed that Deplete Host reads

"After you win a challenge in which DH participated..."

Does this means that DH must still be participating in the challenge at the resolution phase ... or even response phase or does it mean at any time

i.e. there are actions that can stand and remove a participating character but they would still have participated at some point

This is checked at challenge resolution and Depleted Host must be participating at that time.

When a character is removed from a challenge, it's no longer considered to have been participating.

See FAQ entry 3.33:

(3.33) “Participated” and Removal From Challenge
A character is only considered to have
participated if they remain in the challenge
through its resolution. If they are removed
from a challenge, there is no memory of that
character having participated in that challenge.