R: 1. a5-a4, dann 1. Ke8 Ka6 2. Kd8 Lb7 3. Ke8 Lc8 4. Kd8 Kb7 5. Ke8 Ka8 6. Kd8 La6 7. Ke8 Lb7 8. Kd8 Lc8 9. Ke8 Ld7+ 10. Kd8 Le8 11. Kc8 Lf7 12. Kd8 Lg8 13. Ke8 Kb7 14. Kd8 g5 15. Ke8 Kc8 16. a4 Lf7# (z.B)

Firstly: what is the outcome of the game? The only possible mate is from the position with wKc8, sKe8 & wLg8, then wLg8-f7#. sL can squeeze past sK in the corridor, but after that there is a problem:
F: 1. ... Ka6 2. Ke8 Lb7 3. Kd8 Lc8 4. Ke8 Ld7+ 5. Kd8 Le8 6. Kc8 Lf7 7. Kd8 Lg8 9. Ke8 Kc8.
We have reached the pre-mate diagram, but it's Black to move. And there is no way we can shift parity, even using wBg4 for tempo, because in the pre-mate position, there is no prior Black move. So we can't ever reach mate from the diagram. Hence the current position is dead, by Law A1.3.
If Black's last move was 1. Ke8*d8 (with or without capture), that move was forced, and by A1.3 the game would have ended prematurely. We can therefore deduce that Black's last move was 1. a5-a4. This move drew, but had Black instead played the only other option, 1. Kd8-e8, then we now have two spare tempi: (i) for White on the g file, (ii) for either player on the a file.
But now if we try to just march wL past bK, we will find that the parity has changed, and it no longer works:
R: 1. a5-a4, F: 1. Ke8 Ka6 2. Kd8 Lb7 3. Ke8 Lc8 4. Kd8 Ld7=?
We can try to use wK for tempo, but then sK becomes blocked a few moves further on:
R: 1. a5-a4, F: 1. Ke8 Ka6 2. Kd8 Lb7 3. Ke8 Lc8 4. Kd8 Kb7 5. Ke8 Ld7+ 6. Kd8 Le8? 7. KxLe8 forced.
Or we can use the g file tempo:
R: 1. a5-a4, F: 1. Ke8 Ka6 2. Kd8 Lb7 3. Ke8 Lc8 4. Kd8 g5? 5. Ke8 Ld7+ 6. Kd8 Le8 7. Kc8 Lf7 8. Kd8 Lg8 9. Ke8 Kb7 10. Kd8 but to progress we need to spend our last tempo prematurely: 10. ... a4 11. Ke8 Kc8 and again it's Black to play in the mating diagram.
In fact, wL itself must change the parity, by triangulating on the a6-c8 diagonal, e.g.:
R: 1. a5-a4, then e.g. F: 1. Ke8 Ka6 2. Kd8 Lb7 3. Ke8 Lc8 4. Kd8 Kb7 5. Ke8 Ka8 6. Kd8 La6! 7. Ke8 Lb7! 8. Kd8 Lc8! 9. Ke8 Ld7+ 10. Kd8 Le8 11. Kc8 Lf7 12. Kd8 Lg8 13. Ke8 Kb7 14. Kd8 g5! 15. Ke8 Kc8 16. a4! Lf7#
where we used our spare tempi on White's 14th & Black's 16th moves. [Note that triangulation is only possible *before* wL passes sK - when wL reaches g8 it can only vibrate backwards and forwards to h7.]
In conclusion, the alternate game *can* end in mate, confirming that the current position is although dead is reachable.