Dynamically defined covering-systems conjecture

Let $y_1,\dots,y_\ell,z$ be real numbers, with $z$ outside the rational span of $1,y_1,\dots,y_\ell$, and let $\varepsilon,\delta>0$. For any $n_0>0$, there exist positive integers $n_0<n_1<\dots<n_M$ and real numbers $w_1,\dots,w_\ell$. Set

$$K=\prod_{m=1}^M(n_m+1),\qquad k_m=\frac{K}{n_m+1}.$$

Dynamically defined covering-systems conjecture. These choices can satisfy both $\max_i\|n_my_i-w_i\|\leq\varepsilon/2$ for every $m=1,\dots,M$, and

$$\left|\left(\bigcup_{m=1}^M\frac{\mathbb{Z}+z}{k_m}+\left(-\frac{\varepsilon}{2K},\frac{\varepsilon}{2K}\right)\right)\cap[0,1)\right|\geq1-\delta.$$

Here $|\cdot|$ denotes Lebesgue measure and $\|\cdot\|$ distance to the nearest integer. The claim strengthens the preceding typicality heuristic by imposing dynamically defined positive-density choices and an arbitrarily small expected uncovered measure; the supplied text gives no resolution evidence.