I'm looking for an adverb that means "hopefully, but not really" [closed]

You could use wishfully

From dictionary.com

having or showing a wish; desirous; longing.

Maybe not the exact definition you are looking for, but used in the context of the sentences you provided, I think it would work

You could try optimistically

Vocabulary.com says:

An optimistic person thinks the best possible thing will happen, and hopes for it even if it's not likely.

Another possible option is "speculatively".

(1) in a way that is based on guessing or on opinions that have been formed without knowing all the facts; (2) in a way that shows that you are trying to guess something

oxfordlearnersdictionaries.com

vainly:

Courtesy of Google

adverb: in a way that produces no result; to no avail.

Example: She dug vainly into the ground, hoping the pipe was there, even though it really could have been anywhere.

Example: I installed the dubiously-sourced component into the circuit, screwed all the terminals tight, and flipped the switch vainly.

Example: Pat was pretty sure the bank account had gotten zeroed. Still, he vainly shoved the cash machine card into the slot.