The Lemmens-Seidel conjecture for base size $5$

In 2020, Lin and Yu claimed to prove the so-called Lemmens-Seidel conjecture for base size $5$. However, their proof has a gap, and in fact, some set of equiangular lines found by Greaves et al. in 2021 is a counterexample to one of their claims. In this paper, we give a proof of the conjecture for base size $5$. Also, we answer in the negative a question of Greaves et al. in 2021 whether some sets of $57$ equiangular lines with common angle $\arccos (1\mathrm{/}5)$ in dimension $18$ are contained in a unique set of $276$ equiangular lines with common angle $\arccos (1\mathrm{/}5)$ in dimension $23$. In addition, we answer in the negative a question of Cao et al. in 2021 whether a strongly maximal set of equiangular lines with common angle $\arccos (1\mathrm{/}5)$ exists except the set of $276$ equiangular lines with common angle $\arccos (1\mathrm{/}5)$ in dimension $23$.