Uh, no. Mars' gravity is strong enough to hold onto a decent atmosphere by itself, if that atmosphere can be protected from the solar wind. That's the whole idea.
No Vulcans. Phobos stays put. Mars gains an atmosphere and terraforming proceeds as planned. Carl moves to Mars.
The articles that claim a breathable atmosphere can be sustained on Mars depend on a lot of handwaving.... for a long time. And none I’ve read talk about where the nitrogen component will come from, if the atmosphere is not to be essentially O2 plus CO2.