Uniqueness conjecture for minimum universal sets of MacMahon cubes

A minimum universal set is a collection of 12 colored MacMahon cubes that can be used to build all 30 target cubes. The paper has exhibited at least 10 such sets, including Haraguchi's set and additional shifted or symmetric examples.

Uniqueness conjecture. The 10 minimum universal sets are the only collections of 12 cubes that can build all 30 target cubes.

This conjecture asserts that the currently known examples exhaust the minimum universal sets. The statement is presented after computationally finding the additional examples, but no proof or resolution is supplied here.