Consider completing the quest Making Amens! at the same time as it requires visiting the Incinerator room as well. If doing this quest only for the Sleepga II quest, it is wise to do the Never to Return quest at the same time, as both quests have a 3 game day waiting period.
After completing Collect Tarut Cards, have Chululu test your compatibility on three different game days (you don't have to wait an hour; just wait until a new game day).
To test compatibility, input the name of any character within the area as the password. It can be the same person three times. It can also be an NPC (e.g. one of the two standing in front of her stall, herself or Kurou-Morou).
Enter the name exactly as it is in-game, with the first letter capitalized.
Even after you've completed Collect Tarut CardsChululu may still start a dialog with "Oh, what a crying-wying shame!" which launches the quest again. Simply reply "No, Thanks". She will see dark clouds in the horizon, but you'll be able to talk with her again to test your compatibility.
Speak to her on a fourth game day to receive this quest.
To get to the Incinerator, each character who wishes to enter the Crematory Hatch (a door) needs a Garlaige Key (Rare/Ex), which can be obtained from Fallen Evacuees in Garlaige Citadel. There are 3 small rooms on the first map with 2 Evacuees in each.
Once past Gate #2, go to (I-7) on the third map to find the Crematory Hatch that leads into the Incinerator room. Tractoring someone through the door does not work, as Mashira will not respond to you unless you entered the Incinerator room by using a Garlaige key. Everyone in your party must obtain their own key to complete this part.