The torsor compatibility conjecture for period maps

Let $P$ be the algebra of periods and let $T$ be one of the corresponding torsors of isomorphisms, with the period construction giving a map from the period algebra to $T$. In the cases labelled A), B), C), D), and F), the torsor compatibility conjecture says that the map from $P$ to the corresponding torsor of isomorphisms is a map of torsors. The conjecture concerns compatibility of the motivic period construction with the torsor structures; the source gives no resolution status.