Thanks a lot for your answer! May I ask for some clarification on "comparing with the generic fibre"? While taking the generic fiber certainly helps showing faithfulness and the fraction field being algebraically closed helps with essential surjectivity as described above, does it also help with fullness? Is it possible that after the reductions you describe, connectedness is preserved under taking the generic fiber, similar to the case of a discrete valuation ring, see e.g. Tag 055J?