OK, we're going to analyze the different variations to d6 Rd2+:
1. (white exchanges rooks after e7+):
d6 Rd2+ Ke1 e3 d7+ Rxd7 Rxd7 Kxd7 Because of the extra pawn on f7, this is a win for black.
2. (white plays Ra3 after e7+ )
d6 Rd2+ Ke1 e3 d7+ Rxd7 Ra3 f5 This favors black because of a rook pin on the king.
If Bxe3 Bxe3 Rxe3+ Re7, and black wins: Kf2 Rxe3 Kxe3 Ke7 Kf4 Ke6 etc.
If white tries to escape the pin before playing Bxe3 (Kf1) then black can play f4, protecting the pawn on e3. At that point, black has a tremendous advantage with the two connected passed pawns, and we should find a way to win this.
3. (white plays Ra8+ before Ra3 to defend the pin.)
This is a variation on 2, but it's easy to refute:
d6 Rd2+ Ke1 e3 d7+ Rxd7 Ra8+ Rd8 Ra3 and Kd7 allows for Re8 and the pin if white tries to grab the e3 pawn.
4. (white plays Kf1 rather than Ke1 and then Ra3)
So this is one of two variation after Rd2+ that gives white chances, and its the one that worries me and makes me think we might reconsider Rd2+. Two sub variations here: the first one without Ra8+
d6 Rd2+ Kf1 e3 d7+ Rxd7 Ra3 Now because of Kf1 there is no pin. And so:
f5 Bxe3+ Bxe3 Rxe3+ Re7
Obviously white is not going to trade rooks as that is a win for black. But what we need to figure out is, given this board, can we win from this position? (White to move, white with his King on f1, his rook on e3 and a pawn on b4; black with his king on e8, his rook on e7, a pawn on f5 and a pawn on d5.
5. (white plays Kf1, then Ra8+ before Ra3)
This is the other variation that bothers me, and it's probably the most correct for white IF white plays d6: it escapes the pin and forces at LEAST a draw if we trade bishops:
d6 Rd2+ Kf1 e3 d7 Rxd7 Ra8+ Rd8 Ra3 f5 Bxe3
Now we can't play Bxe3 because Rxe3+ and Re5 wins one of our two remaining pawns. The question then becomes, is THIS position winnable for black? (black to move, black's King on e8, his rook on d8, his bishop on g5, his pawns on f5 and b5; white's King on f8, his bishop on e3, his rook on a3, his pawn on b4).
On the face of it, it doesn't look like black can win this. White's pieces are better placed to force a draw here. But keep in mind that to get to this position, white has to play PERFECTLY finding every move, particularly Kf1 after Rd2+ and Ra8+ after Rxd7 before Ra3. These are not easy moves to find unless you're either a grandmaster or you're spending hours looking at this board (like I just have.)
So the question becomes- is this a winnable position for us? And even if it's not, should we take a chance anyhow on Rd2+ figuring that Kaspy won't find this sequence? That's what we need to decide before playing Rd2+ in response to d6.