I don't think its mate in 584 if it's white to play because after 1.Rd8+ Ke2 2.Rd2+ Kf3 (2.Kxe3 3.Rd4+) (2.Ke1 3.Bf2+ Qxf2 4.Rxf2 Kxf2 and there is mate in less than 580) 3.Rf2+ Ke4 (Kg4 Rf4+) 4.Rf4+ Qxf4 5.Bxf4 and mate in less than 579
m a t e i n 5 8 4
Bg4 no more decent checks for black cant defend d8 white is winning
Bg4 is impossible, but working on the assumption of Bg5, stockfish says -0.1
good luck man