Like other pieces, the oldest jars date back to the Song period when trade was established between SE Asia and South China. However, I have seen pieces that look like Han and Warring states with their cloth patterns, for sale. It is difficult for me to say if these are real or not. Back to the point about fakes. Reproductions from Indonesia vary in quality.
Some pieces are obvious.
Some of them are expertly made fakes that would fool the best collectors. When I went back last time I spent many hours shifting in the dealer's shop that I mentioned earlier and some others, simply studying fakes. When I shop I always go to a dealer that simply buys everything - both fake and real. He does not concern whether the items are real or not. He simply sells them at a set mark up for what he originally bought them for.