Continuous-measure recurrence conjecture for recurrence sets

Let $S\subseteq\mathbb Z\setminus\{0\}$ be a set of recurrence. A Borel probability measure $\sigma$ on $\mathbb T$ is continuous if $\sigma(\{a\})=0$ for every $a\in\mathbb T$, and let $e(x)=\exp(2\pi i\tilde{x})$ for $x=\tilde{x}+\mathbb Z$. Continuous-measure recurrence conjecture. There is a continuous probability measure $\sigma$ on $\mathbb T$ such that, for all $m\in\mathbb Z$ and all $\varepsilon>0$, the set

$$\left\{n\in S:\int |e(nx)-e(mx)|\,d\sigma(x)<\varepsilon\right\}$$

is a set of recurrence.

This conjecture is proposed as the remaining ingredient for extending the proof of the main theorem to the question of finding, inside every recurrence set, a recurrence subset that is not a set of strong recurrence. The source gives no resolution.