To make the challange various and find some different ways to have fun with Katamari, I was thinking to play levels for grabbing as many items as I can. I think that it would be a good idea adding this kind of charts for WLK levels (as happend to Katamari Damacy. It already includes # of items charts). All levels can include this charts except of 50 item challange, Save the Pandas, Sumo levels, Save the Earth, Room Cleaning, levels played in As Fast As Possible Mode and Sweet Meadow.
I hope it's ok and someone supports me.

As for the proof, you can take a photo while the king is speaking. If you miss the opportunity, you can replay the level, then your old record (which is the one you have to prove) will appear when the king is speaking (after you finish) above the new one. So, there is no way you cannot get a proof.