Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts

Let $θ\in\mathbb Q\setminus\{0\}$. We construct an approximating function $ψ:\mathbb N\to[0,\frac12)$ for which \[ \sum_{q=1}^{\infty}\frac{φ(q)}{q}ψ(q)=\infty, \] but the set of $x\in[0,1]$ for which \[ |qx-a-θ|<ψ(q),\qquad \gcd(a,q)=1, \] holds for infinitely many $(a,q)\in\mathbb