Shell-wise p-adic Duffin–Schaeffer zero-one law

Let $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$, let $p$ be a prime, and let $k\in\mathbb{N}_0$. Let $\mathcal{C}^p_n$ denote the shell-wise approximation sets and let

$$\mathcal{C}^p=\limsup_{n\to\infty}\mathcal{C}^p_n.$$

Suppose that

$$\operatorname{supp}\psi\subseteq p^k\mathbb{N}\setminus p^{k+1}\mathbb{N}.$$

Shell-wise p-adic Duffin–Schaeffer conjecture. One should have

$$\mu_p\bigl(\mathcal{C}^p\cap p^k\mathbb{Z}_p^\times\bigr)= \begin{cases} (p-1)/p^{k+1} & \text{if }\displaystyle\sum_{n=1}^{\infty}\mu_p(\mathcal{C}^p_n)=\infty,\\ 0 & \text{if }\displaystyle\sum_{n=1}^{\infty}\mu_p(\mathcal{C}^p_n)<\infty. \end{cases}$$

This is proposed as a shell-wise $\mathcal{C}^p$-variant of the p-adic Duffin–Schaeffer theorem. The paper presents it as plausible and does not provide a proof or resolution.