The group-theoretic conjecture on central translates of conjugacy classes

Let $G$ be a group, let $X$ be a subset of $G$, and let $H:=\langle X\rangle\leq G$. Assume that $X$ is a conjugacy class of $H$. Central-translate conjecture. If there exist distinct central elements $z_1,z_2\in Z(H)$ such that

$$z_1x^{-1},z_2x^{-1}\in X$$

for all $x\in X$, then

$$x^2\in Z(H)$$

for all $x\in X$. This is presented as a group-theoretic formulation equivalent to the specialization of the preceding quandle conjecture to subquandles of conjugation quandles; the claim remains open.