Scientists have demonstrated that the Sun orbits around the solar system's centre of mass.
Dr. James O'Donoghue, a planetary scientist at the Japanese space agency, JAXA and who has also worked at NASA, shared his fact-based video on his Twitter. He demonstrated that the centre of mass called the barycenter is the point of an object around which the sun and other planetary objects orbit.
The barycentre is the gravitational centre of the solar system, which has its mass distributed evenly on all sides, and it rarely lines up with the center of the Sun.
Every single object in the solar system exerts a gravitational pull on everything else. The solar system is a massive celestial cluster, and it balances out at a specific point called the barycenter, which is continually changing due to changes in the orbital path of the objects. Everything in the solar system orbits around the barycenter. But the Sun holds 99.87% of all the mass in the solar system, and hence the barycentre will always be near to the Sun.
Technically, it is proven that the Sun and every other object within the solar system orbit around the Solar system's central mass.