If the Sun was a white dwarf, is it possible that it could start orbiting Jupiter?

Since a huge planet (like Jupiter) is massive compared to some white dwarfs, is it possible that if the 2 of them were in one solar system, the dwarf

Since a huge planet (like Jupiter) is massive compared to some white dwarfs, is it possible that if the 2 of them were in one solar system, the dwarf would start orbiting the planet because of the superior gravitational pull?

or:Since a huge planet (like Jupiter) is massive compared to some white dwarfs, is it possible that if the 2 of them were in one solar system, the dwarf would start orbiting the planet because of the superior gravitational pull?

or:Actually, the Sun is stable- it doesn't move. Jupiter orbits the Sun, not the other way around.

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