Sometimes it works, sometimes it doesn't. So I say just scrap this card and rework it to be as simple and bland as humanly possible. Make it what it really is, a sympathy card.
Here we go.
•Passage of Persistence•
This card cannot be flawed. Obtaining 7 wins rewards the weekly Adept Weapon. This card cannot be used to focus an Adept weapon at Saint-14. This card cannot grant access to the lighthouse. This card can only be used once per week per account.
The ye olde 1 & done.
You get the weapon but lose out on everything else that one would get from going flawless. No lighthouse loot, shaders, ships, ghosts, memento, additional adept drops on wins, nada.
I guarantee that in the time it takes for most players to finish this card, the regular core Trials players would've already went flawless 10x over, probably from 5-0ing the poor fools trying to finish this card.
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Mostly agree with you, my only change would be to allow people with this card to either, 1. Be able to reset and do it all over again for more loot. Or 2. Be able to playing and get adept drops on Ws. This would actually keep people playing all weekend long.