How about 34.d7 Rcc2 35.Rc4 Rxc4 36.bxc4 h5 37.Kc1 Rd4 38.Kc2 Kf6 39.Rh7 Kg6!
This seems to just hold the draw for black.
[d] 8/1p1P3R/6k1/p4p1p/2Pr4/5P2/P1K3P1/8 w - - 6 7
1. d7 Rcc2 2. Rc4 Rxc4 3. bxc4 h5 4. Kc1 Rd4 5. Kc2 Kf6 6. Rh7 Kg6 7. Kc3
Rd1 8. d8=R Rxd8 9. Rxb7 Rc8 10. Rb2 (10. Rb6+ Kg5 11. Kd4 Kf4 12. c5 Rd8+
13. Kc3 Rd5 14. Rc6 Rd1 15. Rh6 Rc1+ 16. Kd4 Rd1+ 17. Kc4 Ra1 18. a3 Rxa3
19. c6 Ra2 20. Rg6 Rb2 21. Kd5 Rc2 22. Rg8 a4 23. Ra8 Rd2+ 24. Ke6 Re2+ 25.
Kd5 Rd2+ 26. Ke6 Re2+ 27. Kd6 Rd2+ 28. Kc5 Rc2+ 29. Kb6 Rb2+ 30. Kc7 Ra2
31. Kd6 Rd2+ 32. Ke7 Re2+ 33. Kd7 Rd2+ 34. Kc7 Ra2 35. Kd7 Rd2+ 36. Ke7
Re2+ 37. Kf6 Rc2 38. Rxa4+ Kg3 39. Kg5 Kxg2 40. f4 Rxc6 41. Ra5 h4 42. Kxh4
Rg6 43. Rxf5 Kf3 44. Rf8 Rg4+ 45. Kh3 Rxf4 46. Rxf4+ Kxf4 {Insufficient
material}) 10. .. Kf6 11. Re2 h4 12. Kd4 a4 13. c5 a3 14. Kd5 Rb8 15. Kc4
Rb7 16. c6 Rb6 17. Kd5 Rb2 18. Re6+ Kf7 19. c7 Rd2+ 20. Ke5 Rc2 21. Rh6 Kg7
22. Rxh4 Rxc7 23. Ra4 Kg6 24. Kf4 Rc6 25. Rxa3 Rc4+ 26. Kg3 Kg5 27. Ra8 f4+
28. Kh2 Rc5 29. a4 Kh4 30. Rf8 Kg5 31. Ra8 Kh4 32. Rf8 Kg5 33. Rb8 Kh4 34.
Rb4 Kg5 35. Rb5 Kh4 36. Rb8 Rd5 37. Rb4 Kg5 38. Rb5 Kh4 39. Rb7 Rc5 40.
Rh7+ Kg5 41. Rb7 Kh4 42. Rf7 Kg5 43. Ra7 Kh4 44. Rb7 Rd5 45. Rb4 Kg5 46.
Kh3 Rd2 47. Rb5+ Kf6 48. a5 Ra2 49. Kg4 Ra4 50. Rb6+ Ke5 51. Rg6 Ra2 52.
Rg5+ Kf6 53. Rh5 Ra4 54. Rh6+ Ke5 55. a6 Kd5 56. Rf6 Ke5 57. Rf5+ Ke6 58.
Rh5 Kd6 59. Rh7 Ke5 60. a7 Ra2 61. Re7+ Kd6 62. Rb7 Ke5 63. g3 fxg3 64.
Kxg3 Kf5 1/2-1/2
[d] 8/PR6/8/5k2/8/5PK1/r7/8 w - - 1 65
However I feel that there is a good chance there is a forced win for white somewhere in this line.
Now cracks a noble heart.—Good night, sweet Princess, And flights of angels sing thee to thy rest!