Are both these sentences correct:
a) "I will HOLD UP hope until there is a chance that this could happen."
b) "I will KEEP hope until there is a chance that this could happen."
c) "I will MAINTAIN hope until there is a chance that this could happen."
Those don't work well. I think you're looking for "remain hopeful". I will remain hopeful until ...
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Those don't work well.
I think you're looking for "remain hopeful".
I will remain hopeful until ....
"will have hope" and "will lose hope" and "will inspire hope" are some of the combinations that are often used with "hope", but those are not really suitable in your example sentences. Well, maybe "will have hope", but "will remain hopeful" is better.
CJ