What's interesting about both your solution and my solution is that they share certain structural properties that seem mandatory for minimizing the maxima of the 3-number sums:

1. The five highest numbers (11-15) are spaced exactly 2 elements apart

2. The five lowest numbers (1-5) are spaced exactly 2 elements apart

3. The five middle numbers (6-10) are spaced exactly 2 elements apart

Thus both solutions are 5 triplets of a pattern [low,high,medium] (or it's reverse which is the same).

The cardinality of these patterns is quite modest, being at most (5!)^3 = 1,728,000 for a simple algorithm that tries every possible permutation of the highs, lows, and mediums to being 1/10th that number if one cleverly pre-culls the reverse sequence or congruent rotations of the sequences.

It's a separate issue whether there may be solutions that don't have this structure of five triplets. For example, might something of the form {medium,12,low,11,medium,....} lead to MinOfMaxSum of 25 or lower? Perhaps some pattern of three 5-element subsequences has the same or better MinOfMaxSum.