Variance vanishes but the named mean is wrong

For real $\gamma \ne 0$, take as residual state the coherent vector

Then $n_{ϵ}={\gamma }^2/ϵ$, and after the declared displacement by $z/\sqrt{ϵ}$,

Thus the variance condition alone passes, but the mean does not converge to the named target. This is preparation bias, not a failure of the variance calculation.