According to the newspapers (!!) after the Washington St fire, the CFD revised procedures to limit the number of "assumes command" as higher Chief Officers arrive, using the Incident Command structure. I wondered about that - it sounds like you could have a Battalion Chief running a 5-11and higher chiefs doing support. In the above illustration, who would be the IC? The box/fire attack chief?
For any confirmed high-rise fire, a Still and Box & EMS Plan 1 are automatically requested. A Deputy District Chief is assigned on a Still and Box so no, the fire Attack Chief would not be the Incident Commander (I.C.). They usually want the IC looking at the "big picture" from outside or in the lobby directing operations. I could see the DDC being the IC for the duration, unless some higher ranking Chief Officer comes on the scene and wants to be in charge.