Another Faulty Puzzle

Puzzle number 58109 (lichess.org/training/58109) has a better solution than the one given.

The "correct" puzzle answer is Ne4, Qe7, Nxg5+, fxg5. Thus, white trades his knight for a rook.

Another solution, albeit one that results in "Good move. But you can do better" is Ne4, Qe7, and then H4 trapping the rook. Is this not a better solution?

Never mind, I just now saw Qxc6 which was not included in the puzzle.

TITAN27, after 1.Ne4 Qe7 2.Nxg5+ fxg5 3.Qxc6 is winning a whole rook for white. After 1.Ne4 Qe7 2. h4, Black will reply Rg7 and preserve his rook.

Thanks Thezombieman.