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I didn't spend too much time on this, but I couldn't find a mate after h6. There must be, because black won't sac the queen, but the excercise's answer should be the mate that happens instead of the played move.

that f5 is just a "no move - move", like a6, a5, b6, b5 would have been. But, true is, I don't understand why white doesn't move Rh3 after it, since the only option for black to avoid the mate (thank you for pointing them out, schakertije, it was a lazyness from my part not to find them) is to go on with "waiting-meaningless" moves until sac more material in order to delay the mate a move or so. True is that the given continuation leads to mate, but it takes more moves.
Engine's fail? Very, very unlikely in such a tactic and "mate in few" kind of situation. We could have missed something.
More "human" moves from black like 1... Rd5 also fail to Ng6+.
The alt is interesting: clearly inferior, it bets that to mantain the mate threat, and compelling the black queen to stay anchored to the defence of h7 whorts not to have taken it with the rook.
But white is unable to keep the pressure, and doesn't see a way to escape the black queen checks, so start to defend, and his attack loses steam.

The queen sac shown in the problem just delays the mate two moves, when white would spend one "stupid" tempo just putting the rook back where it was before taking the queen, h3, with the same deadly threat than before. Having no better moves than "f5" (that doesn't solve black's problems) it is really strange the computer doesn't go after Rh3 again.
Maybe this shows the fundamental difference between a human and a computer. The queen's sac doesn't save black, but it is still an option for a machine. Not for a human.
But the mystery is why black goes with Nxf5, a move that throws away the key point of white's mate threat (Ng6+). Something that even a medium rated player would not have done, given he understanded where the strengh of white's position really lied on. But the computer does!

@damalfi Well im sorry to burst your bubble. But I just made a puzzle out of it because I couldn't find the answer also and just thought when I make a puzzle for fastest mate the solution for black should come up. The computer gave this solution for black:

So after
1. ... Qc4
2. Rh3 Qd3
3. Rxd3 f5

4. Rh3 Rd6!
5. Ng6+ Rxg6

And

4. Ng6+ Nxg6
5. Rh3 Nf8!

White can't mate! Computer is correct.