The alternative p-adic condition for adelic Beltrami differentials

Let $p$ be a prime, and let $\frac{dbc}{d{\mathbb Z}_{p}}=0$ denote the vanishing of the derivative of a p-adic Beltrami differential $bc$ along ${\mathbb Z}_{p}$. The renormalized average series condition is the condition used in the definition of an adelic Beltrami differential.

Alternative p-adic condition conjecture. In the p-adic case, the renormalized average series condition in the definition of an adelic Beltrami differential can be replaced by the condition

$$\frac{d\mu}{d{\mathbb Z}_{p}}=0.$$

This replacement is not claimed to make the two conditions equivalent; rather, the whole subsequent theory, in particular the Ahlfors–Bers theorem, should hold under this alternative condition.

The conjecture proposes an alternative analytic hypothesis for the p-adic component of the theory. The source does not state a resolution, so it remains open.