Gap-decay and extremal-frequency conjecture for the critical almost Mathieu operator

Let $p$ and $q$ be coprime integers with $q$ odd. For the periodic problem $\Sigma(p/q)$, let $\gamma_k(p/q)$ denote the length of the gap with label $k$. Gap-decay and extremal-frequency conjecture. There is a constant $R$ satisfying

$$8\ge R>3.05124276000\pm 1.56\cdot 10^{-12}$$

such that

$$\min_k\gamma_k(1/q)>\min_{p,k}\gamma_k(p/q)>\min_k\gamma_k(2/q)>R^{-k}.$$

This conjecture records the numerically observed extremal behavior of the smallest periodic spectral gaps and an exponential lower bound on their decay; it was checked computationally for odd $q\le 95$, while its validity in general remains open.