The problem with a fully AI run company is that it needs reliable measurements to base its decisions on. An AI overlord managing a company would typically have some work to complete, would hire people based on the required qualifications (an API call to a temp agency), and assign tasks to employees based on the work to be done. (ofcourse, it has to figure out the workflows first. But once the AI has figured out what exactly needs to be done, I doubt many humans will be involved)
To manage the work effectively, would require that all assigned tasks can be verified and ranked in quality of work. Otherwise some employees could easily do nothing or a lousy job all day, and just enter into the system that all their work has been done successfully. How would the system know who is slacking? Perhaps if we remove all privacy through computer vision, all kinds of sensors, and massive amounts of data entry. But this would still not be comparable to a manager taking a stroll through the employees and talking to people. I really can't see AI running a company at this time. It requires too much input and still won't be able to make reliable decisions.