The prime-set level conjecture

Let $G=\mathbb{Z}$ and let $X=\{x\}$ be a one-element group, with the level structure described above. For an integer $k$, set

$$P_k=\{p\text{ prime}:p\geq k\}.$$

Prime-set level conjecture. For every integer $k$,

$$-\operatorname{lv}(P_k)=2.$$

This is presented as an assertion about the level map in the elementary example; the supplied text gives no indication that it has been proved or disproved.