According to the Wiki article 株式会社ビーエムジージャパン is what's written in official documents.
Like I said, the english name is the same, but legally they're different entities. And they're still different in the timeline. Logically you can't have
X --> succeeded by Y --> succeeded by X
The final company happens to have the same name, but since it came about due to Y changing its name, it's not the same as X.
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