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What are some relations between the existence of Motivic t-structures and Motivic galois groups?

I heard that indeed the existence of the Motivic t-structure implies the isomorphism between Ayoub's Motivic galois group and Nori's. Is this correct?

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The argument sketched in Example 3.20 of [J. P. Pridham, Tannaka duality for enhanced triangulated categories, arXiv:1309.0637] demonstrates the comparison assuming the existence of the motivic t-structure.

On the other hand, an unconditional proof is given in [Utsav Choudhury, Martin Gallauer Alves de Souza, An isomorphism of motivic Galois groups, arXiv:1410.6104].

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