Common-collector Colpitts oscillator - feedback analysis

I am a student working on a project where we are using CC Colpitts oscillators as local oscillators.

We've chosen the above values based on the condition g_m> w₀²C₁C₂R_L,series, which for the given image translates to g_m> w₀²C₁C₂R₆.

This is a formula we worked with as a given result (assuming biasing resistors are large in comparison to impedances at the resonant frequency), which I gather comes from the feedback network needing to have closed-loop gain > 1 and have the feedback signal be in phase with the input.

I wanted to arrive at this result myself but have been having some trouble. Here is how I have framed the problem and tried solving it:

1. The feedback topology is shunt-shunt, i.e. I am sampling the voltage at the emitter and then feeding a current back in at the base.
2. The feedback gain, B = j(1 / X_C1), since the feedback input comes into one end of C1 and the other end is grounded for feedback analysis (shunt connection).
3. Then for an open-loop gain A, the condition of A*B > 1 should allow me to solve for the previous result.

I then tried finding the loaded forward open-loop gain (A), loading the emitter with C1||C2 and the base with L||C1. However, here I got lost in the math and haven't been able to reduce A*B > 1 to the above result after multiple attempts.

I would appreciate any input of mistakes I may have made in the problem setup or help on how to proceed from here if the rest is right.