I have tried the mechanical solution instead. Insert a flat head screwdriver under the edge of the connector (one at a time) and exert force so that you push the connector away from the PCB:
For me, the solder joint that was holding the side tab to the PCB cracked and the plastic housing lifted slightly on that side. Then I repeated the procedure on the other side with a similar effect. If the solder joint doesn't crack, you may end up lifting the side tabs' PCB pads, but that wouldn't be a concern unless you were planning in soldering the connector back later on (although it seems that it would be strong enough even without the side tabs).
After the plastic housing of the connector is no longer held in place by the two side tabs, you can lift it up and remove it. The final result was this:
pretty clean job, innit?
Now desoldering the 4 legs can be done one at a time and it should be straightforward even without specialised tools.
EDIT:
clearly, make sure you do this with all the required care. You could end up damaging traces if you were to misuse the screwdriver. Check the PCB diagram to see where the traces are and steer clear of them. Also, needless to say, any of these interventions will void your warranty.