I can imagine, that 1.d3 has the edge as for sparing h3 longer, so 50 moves- count resetting can be delayed till necessary. Difference to 1.d4 shows after
1. d4 Kxe1 2. d5 Ke2 3. d6 Nxd6 4. Nxd6 Ke1 5. Nb5 Ke2 and 6.(or 7. transposing with Nd4) h3 is needed already to get Black into zuwgzwang, 6...Ke1 7.Nd4 Kd1 8.Kf1
That makes a difference to
1. d3 Kxe1 2. d4 Ke2 3. d5 Ke1 4. d6 Nxd6 5. Nxd6 Ke2 6. Nb5 Ke1 and Black with 7. Nd4 is already in zugzwang without h3
7...Kd1 8. Kf1
so h3 can be delayed till 50 moves arise for first time.
Yet I gave up with best DTM anyhow, something like this
[pgn][Event "?"]
[Site "Dort"]
[Date "2023.05.10"]
[Round "1"]
[White "Hobacle, Single"]
[Black "Horse, First?"]
[Result "1-0"]
[SetUp "1"]
[FEN "4n2n/5ppr/5prb/1N3Pp1/5pP1/5P2/3Pk1PP/4N1K1 w - - 0 1"]
[PlyCount "149"]
1. d3 Kxe1 2. d4 Kd2 3. Kf1 Kd3 4. d5 Kc4 5. d6 Nxd6 6. Nxd6+ Kd3 7. Ne4 Ke3 8. Nf2 Kd4 9. Ke2 Kc3 10. Ne4+ Kc2 11. Ke1 Kc1 12. Nc5 Kc2 13. Kf2 Kc3 14. Ke2 Kb2 15. Kf1 Kc2 16. Ke1 Kb2 17. Nd3+ Ka3 18. Kf1 Kb3 19. Kf2 Kc4 20. Ke2 Kd5 21. Kd1 Kd4 22. Kd2 Kd5 23. Kc3 Kc6 24. Kc4 Kb6 25. Nb4 Ka5 26. Nd5 Ka4 27. Nb6+ Ka3 28. Kd3 Kb3 29. Nd5 Kb2 30. Kd2 Kb3 31. Ne7 Kb2 32. Nc6 Ka3 33. Kc2 Ka4 34. Nd4 Kb4 35. Kd3 Ka5 36. Kd2 Kb6 37. Kc2 Ka7 38. Kc3 Ka8 39. Kc4 Ka7 40. Kc5 Kb7 41. h3 Kc7 42. Nb5+ Kb7 43. Na3 Kc7 44. Nc4 Kb7 45. Kd6 Ka7 46. Kc7 Ka6 47. Kc6 Ka7 48. Kb5 Kb8 49. Nd6 Kc7 50. Kc5 Kd7 51. Nb5 Kc8 52. Kb6 Kd8 53. Kc6 Kc8 54. Nc7 Kb8 55. Kb6 Kc8 56. Na6 Kd8 57. Kb5 Kd7 58. Kc5 Kd8 59. Kd6 Kc8 60. Kc6 Kd8 61. Nb8 Kc8 62. Nd7 Kd8 63. Nb6 Ke8 64. Kc7 Ke7 65. Nd5+ Ke8 66. Kc8 Kf8 67. Kd7 Kg8 68. Nb6 Kf8 69. Kd8 Kg8 70. g3 Kf8 71. Nd7+ Kg8 72. Ke8 fxg3 73. Nb6 g2 74. Nd5 g1=Q 75. Ne7# 1-0
[/pgn]
won't be fully correct, e.g at the very start 2....Kd2 ?! makes it shorter as for the better way shown before with 2...Ke2, regards