The Orthodox tradition regarding the True Cross gives a clear account that the Cross was deliberately cut up into pieces and distributed throughout the Empire after its capture by the Persians and subsequent recovery. The Church did want a relic of such great importance ever to fall completely into the hands of non-Christians again.
One thing I noticed in the New Advent article you linked to was that the Cross was speculated to be made of pine. An interesting Orthodox Tradition is that it was made up of three different kinds of wood: pine, cedar, and cypress. It alludes to Isaiah 60:13 in the LXX as a prophecy of this.
I had not heard of the work of de Fleury -- I never doubted that the rationastic stories about the relics of the True Cross were gross exaggerations with a definite agenda, but I hadn't learned that anyone had done the work to disprove such tales.
Maybe the wood is the same wood, but how do we know that the wood came the Cross. Like I said, God left us nothing material to fight over, nothing material is equal to God. He left us only His Spirit through His Word.