Maxwell's equations (in differential form) and Jefimenko's equations are in some sense inverses of each other: Maxwell's equations tell you how to use the EM fields $F$ to calculate the current (and charge) density $J$, and Jefimenko's equations tell you how to use the current density $J$ to calculate the EM fields $F$.
Maxwell's equations are only internally consistent if the current is conserved. Put another way: given any EM field $F$, the corresponding current density $J$ that comes out of Maxwell's equations will automatically be conserved.
Q1. What happens if you just blindly plug a non-conserved current $J$ into Jefimenko's equations? Do you still get a seemingly sensible EM field $F$?
Q2. If the answer to Q1 is yes, then what happens if you take that EM field $F$ and plug it into Maxwell's equations to try to get back a current density? The resulting current density $J'$ must be conserved if it comes out of Maxwell's equations, so clearly you can't end up with the same current density $J$ that you started with. This seems to suggest that composing together Maxwell's and Jefimenko's equations gets you some kind of projection map from the space of all possible four-current fields $J$ to the subspace of conserved four-current fields. What is that nature of this projection map? (My guess is that there's some natural decomposition of a general differential form $J$ as the sum of a closed part and a non-closed part, and this projection map simply discards the non-closed part.)