Um, maybe I'm reading you wrong but I highly doubt that Mather is going to demote himself from the role of President & COO. Other than some wishful thoughts from some sports writers about Lincoln, I haven't seen any indication that they are thinking of changes to the executive structure - it's just the 'baseball operations' that Mather is reviewing. As far as I can tell, they are looking for a GM to work within the existing org, not to undertake a serious reorganization.
Whoever they get, the GM will be reporting to the President/COO, Mather. As with JackZ, that GM will have authority over baseball operations, although how much authority is to be determined. There is some indication that the new GM may not have say in whether Lloyd and his staff stick around and if the GM can't make that call, then I fear the next GM will be neutered from day one.