Fisher's zero–one law for the spectral gap

Let $\mu_2$ be the normalized Haar measure on $SU(2)$ and, for $g \in SU(2)^n$, let $z_g$ be the Hecke operator $(z_g f)(x) = \sum_{j=1}^{n}\bigl(f(g_jx) + f(g_j^{-1}x)\bigr)$.

For every $n \ge 3$, exactly one of the following alternatives holds (at least one, formally):

1. for $\mu_2^{\otimes n}$-almost every $g\in SU(2)^n$, $\lambda_1(z_g) < 2n$ (spectral gap);
2. for $\mu_2^{\otimes n}$-almost every $g\in SU(2)^n$, $\lambda_1(z_g) = 2n$ (no spectral gap).

Hence, to prove the spectral gap conjecture for $n\ge 3$ it suffices to find a set of $n$-tuples of positive Haar measure that all have a spectral gap.

Formalization Note. Alternative 2 is stated as "almost every tuple fails the spectral gap property", which is equivalent to $\lambda_1 = 2n$ since $\lambda_1 \le 2n$ always.