Well, 1.c7 will just lose since 1....Rxf2 comes with check, and matter where the king goes, white will get mated on either 1st rank or on the h-file since black can cut off the g-file with a pawn on h5 when necessary. The same applies to 1.cxd4 to get to connected passers- black just mates after 1...Rxf2.
However, white probably also loses with 1.Qxd4 protecting f2, but black now wins the bishop on b8. Here is the best line I can see with this line:
1.Qxd4 Rxb8 2.Qc4+ Re6 3.f4 Kh8.
I just don't see how the pawn can help in this line- black can always retreat both rooks to the 8th rank to protect each other and block the c-pawn. Maybe there is a way to win this as white but I am not seeing it, and once the knight gets activated, black will probably double attack the c-pawn and win it along with the game.
I think white probably needs to look at 1.Kh3 to buy a tempo to be able to play cxd4 creating connected passers, but even this looks bad:
1.Kh3 Rxf2 2.cd4?? Rxh2+ 3.Kg4 Rbf2 and suddenly another mating net is created that I don't see a way out of.
So, this seems to leave...
1.Kh3 Rxf2 2.Qxd4 Rxh2 3.Kg4. Not sure about this yet- black has several replies, and the only one I can clearly calculate is
2......Rxb8 3.c7! Rc8 4.Qd8 looks winning. However, black doesn't have to take the bishop in the first place. I need to think about this a bit more.