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The idea that periods should be subject to a "transcendental Galois theory" has been first advanced by Grothendieck, who sketched a beautiful (but extremely conjectural) relationship with his theory of motives and motivic Galois groups. The resulting Period conjecture is very closely related to the Kontsevich-Zagier period conjecture that you mention in your question. I could give more details, but instead I will recommend this short and beautiful survey together with this other survey for some recent developments.