The higher-order recurrence conjecture for Bohr-Hamming balls

For $r\in\mathbb N$, let $BH:=BH(\bm\beta,\bm y,k,\varepsilon)$ be a proper Bohr-Hamming Ball, with $\varepsilon>0$ and $y\in\mathbb T^r$. Define

$$BH^{1/k}:=\\{n\in\mathbb N:n^k\in BH\\}.$$

A set is $(\delta,k)$-recurrent when it has the $(\delta,k)$-recurrence property used in the paper. Higher-order recurrence conjecture. For all $\delta>0$, there exists $k_0\in\mathbb N$ such that for every $r\in\mathbb N$ and every proper Bohr-Hamming Ball $BH:=BH(\bm\beta,\bm y,k,\varepsilon)$ with $k\geq k_0$, $\varepsilon>0$ and $y\in\mathbb T^r$, the set $BH^{1/k}$ is $(\delta,k)$-recurrent. This is proposed as an analogue of the preceding recurrence lemma and as a possible route toward higher-order recurrence results for $k\geq3$; the source does not indicate whether it is known or resolved.