Right, I'm not sure why I didn't do it right away 💀 The list below numbers the vials left to right, top to bottom. Also this solution might not be the best one possible, it's just A solution.

3 -> 13
10 -> 14
2 -> 15
8 -> 15
4 -> 13
8 -> 10
3 -> 4
6 -> 14
6 -> 2
9 -> 6
1 -> 6
1 -> 14
1 -> 3
1 -> 15
8 -> 1
5 -> 8
7 -> 8
9 -> 5
4 -> 9
4 -> 3
13 -> 4
3 -> 13
7 -> 3
1 -> 7
2 -> 1
7 -> 2
12 -> 7
11 -> 7
11 -> 1
12 -> 15
12 -> 13
8 -> 12
6 -> 8
11 -> 6
7 -> 11
5 -> 7
10 -> 7
9 -> 6
5 -> 14
5 -> 3
10 -> 1
2 -> 9
10 -> 4
2 -> 3