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...I am not sure if it can be mounted on your lens, it cannot be mounted on the 150 macro, but the Sigma tc can.
Yes, the EF Extender 2x III can be mounted on the EX 120-300/2.8 OS.
I posted example photos of this lens with the Extender 1.4x III last year,
Unfortunately, I can't find any examples of the 120-300/2.8 OS with 2x III, and I don't have the lens, anymore.
OTOH, I can say this - IQ from the Extender 2x III is a lot better than from the 2x Mk II.
I figure if the 2x III is better that the Sigma APO 2x, then lowa2 owes me two beers. Afterall, "APO" is supposed to be way better than "III". Otherwise, we're even. He buys the first one.