Nonexistence conjecture for balanced neighborly homology spheres with 6k vertices

Let $k\geq 2$. A balanced $k$-neighborly homology $(2k-1)$-sphere is a balanced simplicial homology sphere of dimension $2k-1$ in which every set of at most $k$ vertices with distinct colors is a face. Nonexistence conjecture. For an arbitrary $k\geq 2$, there does not exist a balanced $k$-neighborly homology $(2k-1)$-sphere with $6k$ vertices. This conjecture concerns the first non-trivial case in the proposed existence problem for balanced $k$-neighborly homology $(2k-1)$-spheres with equally sized color classes; its status is not resolved in the source.