For years, many investors and employees at Space Exploration Technologies have regarded the company’s No. 2 executive, Gwynne Shotwell, as the adult in the room at the boundary-pushing rocket company founded by Elon Musk.

Her acolytes have lauded Shotwell’s ability to translate her visionary boss’s wild ideas into a business that has become the most valuable venture-backed startup in America, one that has won key clients such as NASA and has launched an ambitious satellite-based internet service. Respect for Shotwell has run so deep at the company—better known as SpaceX—that some longtimers have a saying: They work for Gwynne, not Elon.

Lately, though, Shotwell’s handling of employee concerns about Musk’s conduct has put that admiration to the test.