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That looks like a Sieg, I had one before and had the same problem - I'm not saying this is the way to do it, as I don't want you to damage your lathe but I applied a bit of heat to tailstock outer, gripped the taper with some grips and it popped out. Remember to push the tailstock out to its full extent - brutal, I know!I have a similar issue developing in my current tailstock (Warco) and can see a rough ring forming in the tailstock and sometimes have to tap the taper/drill chuck/tailstock die holder/centre out with a piece of wood.....maybe it needs replacing or machining smooth?
On my cheap Chine Lathe you can loosen the Morse taper the same way as mentioned by turning the 'Wheel' a bit further - But that didn't work with my 'Live-Centre' Solved it by making a short plastic extension for the Live-Centre that pushes it out too Could you do something similar - perhaps with a Ring ...?Per
...but not confident with it being such a critical part!
I was lucky that there was a hole in the end of the Live-Centre (instead of the 'missing Tang') .... so that made it easy to turn a tight fitting 'Insert' (made from a broken clothes Hanger) Per