She has done most gym challenges actually!! All of these had led to her becoming the Multiregional Champion (MC. Heh, I'm so smart (?)) even if some of these had to be delayed due to... unexpected events.

She does keep every badge in its original case in her room. Despite her new apathetic feelings regarding competitive battles, her badges are still her pride and joy, so she does her best to keep them shiny.

Briar tried the Galarian Gym Challenge once! However, since battling is not really her calling as a professional career, and seeing how seriously they took battles in Galar, Briar became kind of bored after the third gym and dropped out. So she only has the first three badges of the Galar League, she keeps them still but are probably out of sight, as they don't hold an emotional weight for her

K probably got rid of the badges OTL
Mira keeps hers :) sadie has hers but she didnt make it that far before giving up :'D

None of my OCS ever really did gym challenges, but I feel like they would've hung on to the badges. A nice reminder of their accomplishments.

Yes, most definitely!

Badges are considered trophies of his victory, thus Elio would frame his badges as a memorial - especially when paired with the League Trophy he's won for that specific region, which, in his words, 'completes the set'.