Hi!
I don't have a definite answer, but I believe you are on the right track.
In a piston, oil is not only used for lubrication, a huge job it has to do is the internal cooling of the engine.
So more RPM, more work, more cooling.
In a jet, the oil is also used for fuel heating. So more fuel flow, more heat is transfered from oil to fuel. But then it still does some cooling inside the engine, so I am not sure my answer is correct.
Nic