How about we set a Maximum instead of a Minimum? If we frontload the contest winners with minions and artifacts, we'll exhaust our slots for those card types, necessitating "less flashy" cards in the later rounds?
We could devise a card type ratio in advance. In a 25 card set, we could say: 4 FP characters, 4 minions, 5 conditions, 4 events, 3 possessions, 2 sites, 2 artifacts, 1 wild card. Once the Max is hit, that card type is stricken from play (unless you qualify in your submission that you are using the set's wild card).
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I'd love to try a contest where we play 5-day rounds, but I'm less in favor of pre-announcing several cards at once. Speeding up is one thing, but I want us thinking about the same move at the same time.