If you don't know what that is, it's the PQ from the KMS/MSEA mansion. There's been some speculation that we might get it so I went to MSEA and figured it out. I had done it in KMS before but I didn't really know what was going on.
Anyway, first things first. The PQ is for a party of 4. Levels and classes are irrelevant both in terms of member usefulness and who can get in. It gives no EXP but I'll cover the real prizes at the end. With your party of 4, click on Edmunds to enter. You have 10 minutes to complete the whole thing.
Inside, Edmunds will explain:
Spoiler

Though there are 4 "Jump quest" areas, you don't need to do them all. In fact, the farthest one to the right isn't even a jumping area at all; it's just a direct rope up to the top of the map. At the top is a fake Loki Box. Destroy it and attempt to head through the portal. If it works, go on through. If not, go down and try a different one. Once through, you'll be dumped at the top of the map, where you can either rendezvous with your party members coming from other JQs or go on on your own.
A Jump Quest in the First room

On to room 2:
The second room is quite simple. Run down the hallway holding the up arrow. The up arrow will trigger Mirror Ghosts to attack. Don't get hit by them, or you'll be sent back to the center. As long as you don't stop, there won't be a problem, so just keep going. Eventually, one of the Mirrors will drop a Mirror Piece instead of spawning a ghost. Take the piece and press up on the portal. If you are far from the center, it won't hurt to use those mirror ghosts to your advantage to get back quickly. Anyone can go on to the next stage, not just the leader.
Room 3:
In the third, and final, room, talk to Edmunds again. You will remain here until all your time is consumed. For this reason, it's essential to get through the first two as fast as possible. Talk to Edmunds and take his middle option to get 50 pouches. Set these to a hotkey; it's monster catching time! Go up to the Transparent ghosts all around the area and attack them. They can't be hit by skills, so use regular attacks. You will deal 5 damage to them with every strike, no matter what. Once you get it under half HP, use the pouch to catch the ghost. (It's just like Ariant Coliseum.) After all your pouches are consumed, you can get more from Edmunds.
Spoiler 











Exit room
Once your time runs out, you'll come here. Hand over your ghost pouches to Edmunds for a random prize. So far, I've only gotten 10 Sniper Potions, 1 Wisdom Crystal Ore, 10 unagi, 10 white pots, 10 Ice Cream Sundaes, and 1 Dex Crystal Ore. Not looking so great. The important part, however, is the second option:
Spoiler

As near as I can tell, you get 1 point for every 10 Ghost Pouches you've turned in. It rounds down, as well. 43 pouches only counts for 4 points, for example. The Masked Gentleman's hat is, obviously, the hat that the Masked Gentleman wears.

I don't know the stats off hand.
edit:
checked stats on old KMST thread, it appears to be +4 to all.