Does the Item Collection Quest account for items in previous volumes in .Hack

.in the original .hack, at Volume 4 the Item Collection Quest that was canceled due to the bugs that caused the "server shutdown" comes back but i'm wondering that if you imported your save data between each volume does the quest account for all the weapons you had in those volumes

i know in volume 4 you just need to have the weapon with in your inventory, in a party member's inventory or in storage and it's recorded but i find that by volume 3 your running out of room in storeage for all the weapons/amour/items you've been collecting and sometimes your allies get rid of the weaker equipment they keep