My analysis so far showing that 1... Nf4+ is indeed forced, that the YKW variation is losing for black, and that 2... g2 gives black a drawing chance.
Code:
1. e7 Nf4+ (1... a3 2. Qxb6+ Ka4 3. Qa6#) (1... b5 2. Qb6#) (1... c4+ 2. Kd2
Qh6+ 3. Qxh6 a3 4. Qxb6+ Ka4 5. e8=Q#) (1... Ka6 2. Qxb6#) (1... Kb5 2. Qxb6#)
(1... c4+ 2. Kd2 Qh6+ 3. Qxh6 a3 4. Qxb6+ Ka4 5. e8=Q#) (1... Ne1+ 2. Kd2 Nf3+
3. Ke3 Qh6+ 4. Qxh6 a3 5. Qxb6+ Ka4 6. e8=Q#) (1... Ne3 2. Qxb6#) (1... Nh4 2.
Qxb6#) (1... h1=Q 2. Qxb6#) (1... Qh4 2. Qxb6#) (1... Qh5 2. Qxb6#) (1... Qh6
2. Qxh6 Nf4+ 3. Qxf4 h1=Q 4. Qxa4+ Kxa4 5. e8=Q+ Ka5 6. Qa8+ Kb5 7. c4#) (1...
Qh7+ 2. Kd2 Qc2+ 3. Kxc2 Ne3+ 4. Nxe3 h1=Q 5. Nc4+ Kb5 6. e8=Q+ Qc6 7. Qexc6+
Ka6 8. Qxb6#) (1... Qh8 2. Qxb6#) (1... Qg4 2. Qxb6#) (1... Qf5+ 2. Qxf5 h1=R (
2... h1=Q 3. e8=Q Qb1+) 3. b4+ axb3 4. e8=Q Rh8 5. Qf6 Ne1+ 6. Kc4 Rh4+ 7. Qxh4
bxa2 8. Qa8#) (1... Qe6 2. Qxe6 Nf4+ 3. Nxf4 h1=Q 4. e8=Q Qb1+ 5. Ke3 Qc1+ 6.
Kf3 Qh1+ 7. Ng2 Qf1+ 8. Kxg3 Qf8 9. Qxf8 a3 10. Qd7 axb2 11. Qa8#) (1... Qd7 2.
Qxb6#) (1... Qc8 2. Qxb6#)
2. Qxf4 g2+ (2... Qh7+ $2 3. Kd2 h1=Q (3... Qxe7 4.
b4+ cxb4 5. cxb4+ Ka6 6. Nxe7 h1=Q) 4. Qxa4+ Kxa4 5. e8=Q+ Ka5 6. Qa8+ Kb5 7.
Nc7+ Qxc7 8. a4+ Kc4 9. Qg8+ Qd5+ 10. Kc2 Qxg8 11. e4) 3. Kd2 Qd7 4. e8=Q Qxe8
5. Qxh2 Qg6 (5... Qg8 $2 $16 {1.25/6}) 6. Qg1 Qh6+ 7. Ne3 Qd6+ 8. Kc2 Qg6+ 9.
Kc1 Qh6 10. Qxg2 Qxe3+ 11. Kc2 (11. Kd1 b5 (11... Qe6 12. a3 (12. Qa8+ Kb5 13.
a3 Qd6+ 14. Kc2 $14 {0.53/6}) (12. Kc2 Qxa2 13. Qa8+ Kb5 14. Qe8+ $11 {0.00/6})
12... Ka6 13. Kc2 Qb3+ 14. Kc1 $14 {0.70/6}) (11... Kb5 12. Qd5 (12. Qg4 Ka5
13. Kc2 b5 14. Qf3 $14 {0.60/6}) (12. a3 Qe6 13. Qf3 Ka6 14. e4 $14 {0.69/6})
12... Qf4 13. Kc2 Qg4 14. Qf3 $16 {0.72/6}) (11... Qe8 12. Qd5 (12. Kc2 b5 13.
e4 Qe5 14. a3 $14 {0.58/6}) (12. a3 Qf7 13. Kc2 Qb3+ 14. Kc1 $14 {0.70/6})
12... Qg6 13. b4+ axb3 14. axb3 $16 {0.80/6}) (11... Qe7 12. Qa8+ (12. Qd5 Qh7
13. Kc1 Qh6+ 14. Kc2 $14 {0.70/6}) (12. Kc2 b5 13. Qa8+ Kb6 14. e4 $14 {0.70/6}
) 12... Kb5 13. b3 Qd7+ 14. Kc2 $18 {3.07/6}) 12. Qd5 (12. Qg7 Kb6 13. Qg6+ Ka5
14. Qd6 $14 {0.52/6}) (12. Qb7 c4 13. Qf3 Qe5 14. Kc2 $14 {0.57/6}) 12... Qf2
13. Qd8+ Ka6 14. Qc8+ $14 {0.67/6}) 11... Qe7 (11... Qe8 12. Qf3 (12. Qg4 Qe7
13. Qf3 Qe6 14. Kb1 $14 {0.68/6}) (12. e4 Qe6 13. a3 Qb3+ 14. Kc1 $14 {0.70/6})
12... b5 13. e4 Qg6 14. a3 $16 {0.80/6}) (11... Qe6 12. b3 (12. a3 b5 13. e4
Qb3+ 14. Kc1 $14 {0.70/6}) (12. Kb1 Qd7 13. Qa8+ Kb5 14. Qh1 $14 {0.60/6})
12... axb3+ 13. axb3 Kb5 14. e4 $16 {0.83/6}) (11... Kb5 12. b3 (12. Qf3 Qe6
13. Kb1 Qg8 14. a3 $16 {0.76/6}) (12. Qg4 Ka5 13. Qf3 Qe6 14. a3 $14 {0.70/6})
12... axb3+ 13. axb3 Qe6 14. e4 $16 {0.83/6}) (11... b5 12. Kb1 (12. Qf3 Qe6
13. Kb1 Kb6 14. e4 $14 {0.61/6}) (12. Qa8+ Kb6 13. Qd8+ Kc6 14. Qf6+ $14 {0.60/
6}) 12... b4 13. Qa8+ Kb5 14. Qb7+ $16 {0.85/6}) 12. Qa8+ (12. b3 b5 13. Qa8+
Kb6 14. e4 $14 {0.61/6}) (12. Qf3 Qe6 13. a3 Qb3+ 14. Kc1 $16 {0.71/6}) (12. a3
) (12. e4) 12... Kb5 13. c4+ Kxc4 14. Qxa4+ $16 {0.79/6} *
FEN at the end of the interesting line that gives black a drawing chance
8/8/1p6/k1p5/p7/2P1q3/PP2P1Q1/2K5 w - - 0 11
Diagram