Barrett and Jarvis's conjecture on the small eigenvalues of $C_n$

Let $C_n$ be the matrix discussed above, with two large real eigenvalues and $\lfloor \log_2 n\rfloor-1$ remaining small nontrivial eigenvalues. For a small nontrivial eigenvalue, write it as $\lambda$. Barrett and Jarvis's conjecture. The small nontrivial eigenvalues $\lambda$ of $C_n$ satisfy

1. $|\lambda|<1$;
2. $\operatorname{Re}(\lambda)<1$.

This two-part conjecture was based on numerical evidence for values of $n$ as large as $10^6$. The preceding bound $|\lambda|<\log_{2-\epsilon}n$ is weaker, and the conjectured uniform bounds on the small nontrivial eigenvalues remain unresolved here.