Alright I am still trying to wrap my head around people trying to think that the Admirals wouldn't team-up to take out Jack. They would obviously know who the hell he is, and who he works under so there is no chance in hell they would go easy on him. Even if they were on different ships, the formation the ships were in, the two could have easily gotten to one another's ship with little to no difficulty, especially Fujitora.
Now also considering that dog guy (his name escapes me atm), if I recall correctly when he was talking about Jack, said they had read or heard about his Obituary now that generally means the sum ***** is dead. But as common law dictates, no ones considered dead till you see the body. So the Marines/World Government could easily say Jack is dead in a botched attempt to free Doflamingo, but in reality could have captured him in an attempt to lure Kaido out.