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

Let $C$ be the middle-third Cantor set and let

be its Cantor\Lebesgue measure. For a function $:\to[0,\infty)$, define

W_2:=\left{x\in[0,1]: |2^nx|<(n)\text{ for infinitely many }n\in\right}.

$$**Velani's conjecture.** If$$

is monotonic, then

$$(W_2)=\begin{cases}0&\text{if }\sum_{n=1}^{\infty}(n)<\infty,\\1&\text{if }\sum_{n=1}^{\infty}(n)=\infty.\end{cases}$$

This is a Khintchine-type zero\one law for dyadic shrinking targets on the Cantor set. The conjecture is widely open; existing results establish only restricted convergence and divergence cases.