Finite-family conjecture for $(4,2)$ pillars

Let an equiangular set have angle $\alpha=\frac{1}{5}$ and base size $K=4$, and consider a $(4,2)$ pillar and its Seidel graph. Finite-family conjecture for $(4,2)$ pillars. In the case where $\alpha = \frac{1}{5}$ and base size $K=4$, there are only a finite number of families of connected graphs $S_i$'s such that the connected components of the Seidel graph of any $(4,2)$ pillar in an equiangular set is either a graph or a subgraph of a graph in $S_i$. This conjecture concerns the remaining $(4,2)$-pillar case in the Lemmens–Seidel problem; the supplied text gives no proof or disproof.