Maxwell's equations are \begin{align} \nabla\cdot\mathbf{E} & = \frac{\rho}{\epsilon_0} & \nabla\times\mathbf{B} &= \mu_0\epsilon_0 \frac{\partial\mathbf{E}}{\partial t} + \mu_0 \mathbf{J} \\ \nabla\cdot\mathbf{B} & = 0 & \nabla\times\mathbf{E} &=- \frac{\partial \mathbf{B}}{\partial t}. \end{align} Viewed in light of Helmholtz decomposition these equations can be viewed as fixing independent parts of the fields, with the $\nabla\cdot$ column fixing the divergent (irrotational) parts of $\mathbf{E}$ and $\mathbf{B}$ and the $\nabla\times$ equations fixing the solenoidal (divergenceless) parts.
As suggested by the formula for Helmholtz decomposition, the divergent part of $\mathbf{E}$ is given by $$\mathbf{E}_{\mathrm{div}}(\mathbf{x},t) = \frac{1}{4\pi\epsilon_0}\int \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} \rho(\mathbf{x}',t) \operatorname{d}^3\mathbf{x}' \tag1$$ even when $\rho$ depends on $t$, as written above.
The statement in $(1)$ seems to violate causality. It shouldn't matter since $\mathbf{E}_{\mathrm{sol}}$ should also violate causality in such a way as to make the total electric field obey causality. My question is: what are the details that show how the acausal parts of the Helmholtz decomposed parts of $\mathbf{E}$ and $\mathbf{B}$ cancel (esp. does it require charge conservation)?
Put another way, starting from these equations \begin{align} \mathbf{E}_{\mathrm{div}}(\mathbf{x},t) & = \frac{1}{4\pi\epsilon_0}\int \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} \rho(\mathbf{x}',t) \operatorname{d}^3\mathbf{x}' \tag2 \\ \mathbf{E}_{\mathrm{sol}}(\mathbf{x},t) & = \frac{1}{4\pi} \int \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} \times \left(\frac{\partial \mathbf{B}(\mathbf{x}',t)}{\partial t}\right) \operatorname{d}^3 x' \tag3\\ \mathbf{B}_{\mathrm{sol}}(\mathbf{x},t) & = - \frac{1}{4\pi} \int \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} \times \left(\mu_0\epsilon_0\frac{\partial \mathbf{E}(\mathbf{x}',t)}{\partial t} + \mu_0 \mathbf{J}(\mathbf{x}',t)\right) \operatorname{d}^3 x' \tag4 \end{align} what is the process of transitioning to a manifestly causal form of $\mathbf{E}$ and $\mathbf{B}$ (e.g. Jefimenko's equations), and what parts of the two parts of $\mathbf{E}$ above cancel out in the process?