Short answer:
Yes.
Explanation:
There is a significant list of previous CoE digests that contradict ICE digests (thanks CDavis7M for help with identifying these issues). CoE #43 ruled that the “unique” keyword on The White Tree is not ignored when playing it with Wizard’s Trove, although no reason is presented:
However, Wizard’s Trove’s text states that The White Tree’s text is ignored, which would include the “unique” keyword. And ICE #119 explicitly ruled so:QUERY 16
If a Fallen-wizard player plays The White Tree with Wizard's Trove, and the text of The White Tree is ignored, does that mean another player can play The White Tree? Is the “Unique” keyword ignored on The White Tree if Wizard's Trove is used to play it?
No.
As this committee sees no reason to overrule Ichabod’s ruling on this issue, we have decided to revert the previous CoE ruling and instead adhere to Ichabod’s more obvious reading, which is that The White Tree’s text is ignored completely when playing it along with Wizard’s Trove (including the “unique” keyword).12-08-1998 NetRep: Craig Ichabod O'Brien Source
Wizard's Trove states: “ignore card text” when refering to The White Tree. So, for Fallen-wizards, The White Tree is not unique, since the keyword “unique” is part of the card text. Which of the following three examples are legal:
1. A FW player has The White Tree in play, another FW player plays one himself.
2. A FW player has The White Tree in play, a Hero-player plays one himself.
3. A Hero-player has The White Tree in play, a FW player plays one himself.
All three, assuming the FW is playing The White Tree with Wizard’s Trove in all three cases.
Note that this ruling overturns CoE Digest #43 Q16 and CoE Digest #65 Q5.