Construct an integral filtration interpolating the mixed-characteristic Hofstadter regime

Construct an integral filtration over the mixed-characteristic valuation ring $\mathcal O_K$ that interpolates generic-fiber Azumaya descent for a quantum torus at primitive $p^s$-th-root-of-unity flux with the commutative special-fiber degeneration at $p$.

Background

For quantum tori with flux of order prime to $p$, the paper proves an Azumaya description and obtains an étale-local cyclotomic comparison. The authors distinguish this from the mixed-characteristic case of a primitive $p^s$-th root of unity, where the generic fiber is Azumaya after inverting $p$ but the special fiber becomes commutative.

The desired filtration would capture this degeneration integrally over $\mathcal O_K$, rather than only treating the generic fiber or the special fiber separately. The authors explicitly designate this interpolation as an open problem.