Diagram E1: Crucial Choice for Black.
This file shows some White winning cases that are not in the Kasparov-Alterman file, and some cases in which White doesn't win. [Status: still being filled in.]
The bust is 3...Ka3? 4. Kf8 Qf5+ 5. Ke8 Qg6+ 6. Qf7 Qc6+ 7. Qd7 Qg6+ (Black's idea is that 8. Kd8 d3 is not "Hogtie" but holding, but White has a clever "switchback" that displaces Black's Queen) 8. Kf8! Qf6+ 9. Qf7 Qd8+ 10. Qe8 Qf6+ 11. Kg8! Now it seems that Black must play 11...Kb2 in order to escape Qf8+, but then simply 12. Qb5+ Ka1/a2/a3 13. Qa5+! and 14. Kh7 wins. (So please, no more ...Ka3 moves.:-)
This shows that Qd7 is in general a winning interposition, in every case except Black's Queen checking from d5 and King not on the a-file. White plays 7. Qd7! Qb8+ 8. Ke7 Qe5+ 9. Kf7. Lots more to fill in here...
When Black's King is on b2 the danger is maximum for Black, since a Qf6 interposition will win. The only holding cases are Black Qe6 checking Kh3, when Qg4 Qh6+! is a perpetual check, and Black Qf4 checking Kh4, likewise owing to ...Qh6+.
Put Black's King on a1 or even a2, however, and almost all the danger goes away.
Hence in particular in the following position:
Diagram E1: Crucial Choice for Black.
after 1. Qg1+ Black must play...Ka2!, when entertaining play results: 2. Kh7 Qh4+ 3. Kg6 Qe4+ 4. Kg5 Qd5+(!) 5. Kh6 Qc6+! 6. Qg6 Qc1+ 7. Qg5 Qc6+ 8. Kh5 Qh1+! 9. Kg6 Qc6+, and we have seen this before.
We suspect that the general answer is that with g7 vs. d4 any interposition in the punt-shaped region comprising the diagonals d7-e6-f5, d6-e5-f4, and c6-d5-e4 virtually assures a win. If Black's King is on b2 then interpositions on e7, f6, or g5 also win unless Black can take immediate action, and if White's King is on h6 or h5, a Qg6 interposition likewise wins. If White's King is on e8 and Black checks on g6 or h5 then a Qf7 interposition also wins. The interposition Qf3 is ineffective if Black can reply ...Qg6+ with White's King on g2 or...Qg5+ with White's King on g3, and Qg4 is ineffective in cases we show here.