on (one's) say-so
With or according to one's authorization or permission. Although I'm the manager of this department, I can only make changes to the way things run on the general manager's say-so. A: "We're going to start transferring the funds into the new account." B: "On whose say-so?"
on somebody’s ˈsay-so
based on a statement that somebody makes without giving any proof: He hired and fired people on his partner’s say-so.