The shrinking-target conjecture for the doubling map

Let $T_2:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ be the doubling map, let $\mu$ denote Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, let $\psi:\mathbb{N}\to[0,\infty)$, and let $\mathbf{x}\in(\mathbb{R}/\mathbb{Z})^{\mathbb{N}}$. Write $W_2(\psi,\mathbf{x})$ for the set of points $x$ such that $d(T_2^n(x),x_n)\leq\psi(n)$ for infinitely many $n\in\mathbb{N}$. The shrinking-target conjecture.

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

This conjecture seeks the analogue for the shrinking-target problem of the corresponding Borel–Cantelli law for the doubling map. The supplied text presents it as a natural conjecture motivated by the theorem of Host, but gives no resolution status.