Velani's dyadic approximation conjecture for the middle-third Cantor set

Let $\psi:{\mathbb{N}}\to{\mathbb{R}}^+$ be monotonic, and let

$$W_2(\psi)=\left\{x\in[0,1]:\left|x-\frac{p}{q}\right|<\frac{\psi(q)}{q}\text{ for infinitely many }(p,q)\in{\mathbb{Z}}\times\mathcal A(2)\right\},$$

where $\mathcal A(2)=\{2^n:n=0,1,2,\ldots\}$ and $\mu$ denotes Lebesgue measure. Velani's conjecture.

$$\mu(W_2(\psi))=\begin{cases}0&\text{if }\displaystyle\sum_{n=1}^{\infty}\psi(2^n)<\infty,\\[4pt]1&\text{if }\displaystyle\sum_{n=1}^{\infty}\psi(2^n)=\infty. \end{cases}$$

This is the expected zero--one law for dyadic rational approximation in the middle-third Cantor set, analogous to the known triadic result. The source describes it as an expected eventual truth and notes that the present work supports it; no resolution is supplied here.