Weak Duffin–Schaeffer conjecture with a moving target

Let $\rho: \mathbb N\to[0,\a0\infty)$ be an arbitrary function satisfying

$$\sum_{q\in\mathbb N}\left(\frac{\varphi(q)\rho(q)}{q}\right)^2=\infty,$$

and let $\mathbf y=(\mathbf y_q)_{q\in\mathbb N}$ be an arbitrary sequence in $\mathbb R^2$. Write $W_2'(\mathbf y,\rho)$ for the set of $\mathbf x\in[0,1]^2$ satisfying the corresponding infinitely-often inhomogeneous approximation inequalities with coordinatewise coprimality, and let $\lambda_2$ denote Lebesgue measure.

Weak Duffin–Schaeffer conjecture with a moving target. For every such $\rho$ and $\mathbf y$,

$$\lambda_2\left(W_2'(\mathbf y,\rho)\right)=1.$$

This is the conjectured two-dimensional moving-target analogue of the inhomogeneous Duffin–Schaeffer theorem. The paper presents it as open; a preceding counterexample for a moving target does not disprove this weak formulation.