7.Rxh3!?

1.Be2? gxh3
2.Ra4 d1=Q
3.Bxd1 Nxd1+
4.Kc1 Nf2
5.Rh4 Nd6
6.Kd2 Nf5

White can exchange the rook at h3 for the pawn. I know you probably already know that the ending is lost in that case, but it is the main line in 1.Be2, and why I mentioned the Nalimov Tablebase.

Bxe2 is a mistake because it allows to win a critical tempo for black.

1.Be2? gxh3
2.Ra4 d1=Q
3.Bxd1 Nxd1+
4.Kb3 Nf2
5.Rh4 Nd6
6.Kc3 Nf5
7.Rf4 h2
8.Rxf5 Ne4+
9.Kd3 h1=Q
It's a win for black.

Now explain why 1.Be2 fails to draw.

V-2

1.Bd3+ Nxd3+
2.Kc2 gxh3
3.Kxd2 Nc1
4.Ra4 Nd4
5.Ra7+ Kxh6
6.Rf7 h2
7.Rf6+ Kh7
8.Rf7+ Kh8
9.Rf8+ Kg7
10.Rf1 Nf3+
11.Ke3 Ng1
12.Kf2 h1=Q
13.Rxg1+ Qxg1+
14.Kxg1
It's a draw.

V-3

11.Ke3 Ne5
12.Rh1 Ng4+
13.Kf3 Nf6
14.Rxh2
It's a draw.

What about 2.......gxh3?

Yancey, I'm so glad you are back!

1.Bd3+ Nxd3+
2.Kc2 Nb2
3.Kxd2 Nc4+
4.Kd3 Nxa5
5.hxg4 Kxh6
It's a draw.