Thouless-type spectral-measure inequality for the critical almost Mathieu operator

Let $p$ and $q$ be coprime integers with $q$ odd, and let $\Sigma_{p/q}$ denote the periodic almost Mathieu spectrum. Let $C=0.915\ldots$ be Catalan's constant. Thouless-type spectral-measure conjecture. For $p\ne1,q-1$,

$$\frac{32C}{\pi}>q\cdot|\Sigma_{p/q}|>q\cdot|\Sigma_{1/q}|>\sqrt{\frac{2}{q}}.$$

The inequalities refine the known asymptotic relation between the measure of rational-frequency spectra and rational approximation to an irrational frequency. They were verified computationally for $q\le95$, but are not established in general.