Velani's lacunary Khinchine convergence conjecture for the middle-third Cantor measure

Let $\mu$ be the middle-third Cantor measure, and let $\operatorname{supp}(\mu)$ denote its support. Let $\psi$ be an approximation function such that the sequence $\{\psi(2^n):n\geq1\}$ is non-increasing.

Velani's conjecture. If

$$\sum_{n}\psi(2^n)<\infty,$$

then for $\mu$-almost every $x\in\operatorname{supp}(\mu)$, the inequality

$$\lVert 2^n x\rVert<\psi(2^n)$$

holds at most finitely often.

This is a lacunary convergence analogue of Khinchine's theorem without requiring monotonicity of the approximation function at all integers. The source attributes it to Velani and supplies no resolution.