Now I am back. I was a bit busy so I could not answer. Anyway, I don't know what happened, why replying was not possible earlier.

I didn't want to pull it apart before I was through the discussions with the seller, but now we made an agreement so I am good to go on. Basically, the seller admitted that the MT2 live center is not normal and we agreed on my proposal refund which basically covers the purchase of a new live center. He could not ship one from EU, and I did not want him to ship from China, which is why we agreed on refund so I can buy from someone else. For the chuck play, he basically said that that's normal, "you get what you pay for", which I understand very well also, even though I was hoping for something better and it is actually outside his own specifications.Anyway, with the refund I paid 250 USD for the 4-jaw 100mm 4-th axis and the MT2 tailstock with the damaged center pin, which I think is still usable, unless I really need the very sharp pinpoint edge. I think that's a good price.

What I did so far to improve the situation is that I removed the chuck and the pulley and cleaned up everything. I reassembled the pulley and also moved the stepper pulley out a bit. After this it's a bit better, there is not as much play in the pulley as before, but of course, the chuck is the important part.

Before reassembling the chuck, I made some measurements on the chuck plate, and that shows no real problem as far as I can judge, so that's good. The thing I noticed was that it was very hard to remove the chuck, and the reason for that is because the chuck plate inner flange (I don't know if that's the proper name of it) is I think a bit too large in diameter, so the chuck is VERY tightly fitted on that flange, I think it is too tight because there is ZERO possibility to adjust the chuck. I will see if I will try to do something about that later, but for now I don't have time, so I just reassembled the chuck after cleaning the surfaces and made some new measurements. To my surprise, the play is now half of what it was before, so it is definitely within the specifications, which means that I am satisfied for now.

So for now, I am OK with this as it is, I will test it ot for some real work also before I try to fix anything more. Perhaps one thing I will do is that I remove the chuck once again and try to clean that flange with some very fine sand paper, maybe some grit or something else will be removed by doing this which might improve it even more. I feel that before staring to grind anything I will need to learn some more about using the 4-th axis. I don't want to destroy it or make it worse, after all, a slow start with a less then perfect chuck is better than a destroyed and useless one.