Bellaterra graphs are two-sided expanders

Let $B_n$ be the $3$-regular Bellaterra graph at level $n$, for $n\geq 1$. A family of $3$-regular graphs is two-sided expanding when there is an $\varepsilon>0$ such that its second-largest and smallest eigenvalues satisfy

$$\lambda_2<3-\varepsilon\qquad\text{and}\qquad\lambda_n>-3+\varepsilon.$$

Bellaterra expander conjecture. The Bellaterra graphs $\{B_i\}_{i=1}^\infty$ are a family of two-sided expanders. The authors' computations suggest an eigenvalue gap of roughly $0.05$, but the conjecture remains unproved in the supplied text.