Large-pillar conjecture for equiangular sets

Let an equiangular set have angle $\alpha$ and base size $K$, and let a pillar be one of the structures in the paper's pillar decomposition. Large-pillar conjecture. There is a constant $C$ that depends on the angle $\alpha$ and the base size $K$, but not to the dimension or rank, of any equiangular set, such that there could not be two pillars of size at least $C$. The conjecture is motivated by the authors' experiments and by asymptotic results suggesting that large equiangular sets concentrate in one pillar; the supplied text gives no resolution.