I am now interested in Broué's abelian defect conjecture and I have read many papers concerning it. For a prime $p$, I informally define a finite group to be a $p$-ATI-group if it has abelian Sylow $p$-subgroups with the trivial intersection property. As is well-known, $p$-ATI-cases are important examples for which there are natural stable equivalences between the module categories of a $p$-block and its Brauer correspondent. It seems that the conjecture has been verified for simple $p$-ATI groups. While Andrei Marcus's technique of reduction to simple groups which is based on the structure theorem for finite groups with abelian Sylow p-subgroups by Fong and Harris just deals with principal blocks. I am not sure if there is a similar method for nonprincipal $p$-blocks.
The following are my two related questions:
Is the conjecture completely solved for principal p-blocks of p-ATI-groups?
Is there any progress in the direction for nonprincipal p-blocks of p-ATI-groups?
