Khintchine's theorem on fibers

Let $W(2,\psi)$ denote the set of simultaneously $\psi$-well-approximable points in $\mathbb{R}^2$, let ${\rm L}_\alpha$ be the vertical line through $(\alpha,0)$, and let $m_1$ be one-dimensional Lebesgue measure on that line.

Fiber Khintchine conjecture. If $\psi\colon\mathbb{N}\to\mathbb{R}^+$ is monotonic and $\alpha\in\mathbb{R}$, then

$$m_1(W(2,\psi)\cap{\rm L}_\alpha)=1$$

whenever

$$\sum_{q=1}^\infty\psi(q)^2=\infty.$$

The source explains that the convergent counterpart fails for rational fibers, while this divergent statement is claimed there and is a fiberwise strengthening of Khintchine's theorem.