Does Coulomb's law,
$$\textbf{E}\left(\textbf{r}\right) = \frac{1}{4\pi\epsilon_0}\int \rho\left(\textbf{r}'\right)\frac{\textbf{r} - \textbf{r}'}{\left|\textbf{r} - \textbf{r}'\right|^3}dV',$$
hold always when $\dot{\rho} = 0$? Even if $\dot{\textbf{J}} \neq \boldsymbol{0}$, as in, for example, a scenario with a spinning ring of charge that keeps speeding up? It seems to me that Gauss's law implies that Coulomb's law always holds when $\dot{\rho} = 0$, but $\dot{\textbf{J}} \neq \boldsymbol{0}$ implies a time-varying magnetic field which alarmingly means that $\textbf{E}$ has a nonzero curl.