Macroscopic field is space-time average of the microscopic fields?

Panofsky and Phillips’ Classical Electricity and Magnetism. Chapter 2, Section 3.

The authors show, assuming the microscopic field $mathbf{mathcal E}$ and the microscopic charge density $rho_a$ are related by $nablacdotmathbf{mathcal E} = rho_a/varepsilon_0$, that the macroscopic field $mathbf E$ is related by
$$
nablacdotmathbf E = nablacdotmathbf{bar{mathcal E}}tag{*},
$$
where bar denotes space-time average.

They then claim that this implies that $mathbf{E} = bar{mathcal E}$.

Question: I don’t agree that (*) implies $mathbf{E} = bar{mathcal E}$. What if the curls don’t match? There can be many functions (which don’t differ by just constants) with same divergence, no?

Note that the microscopic fields are not static.