Given the Extraction Procedure unter 3.6 in the Paper i would try the following Procedure:
1. I would transfer the supernatant into a rotary evaporator. (tert-butyl methyl ether boils at around 55°C so we might need a vacuum to boil it of)
2. To the inlet of the evaporator i would conect a nitrogen flask or a ballon filled with nitrogen and flush the system. (Note that the outlet must be unblocked for this to be safe)
3. I would pull a light vakuum to about 0,5 atm. Then open the inlet (still conected to the nitrogen filled ballon) again and let nitrogen back in the system. After the pressure has reached approx. 1 atm i would repeat this step 2-3 times.
4. After the evaporator is fully filled with nitrogen i would heat the supernatant to 45°C in the water bath and then slowly start to pull a light vacuum till the liquid starts to boil.
5. When all the liquid has distilled over let the pressure normalise again and contiue with adding the mobile phase as described in the paper.
That would be my initial approach. I hope that this might help you. Let me know if it works, i am kind of interested myself.