I think I did something right (or at least nothing wrong) so far:
Officially claiming to check on any signs of rot, but honestly I was just curious if this all works. So I removed the soil around all 5 pieces of the CaliGold corm and checked. The one piece, that I identified before as the most promising, is just pushing a shoot from the fraction as if it was still the whole corm. That tiny little white thing on the tip of the eye has been grown since I replanted it after cutting it - just four days ago!
By the way: I wasn't only curious. I really did spray some fungicide :-)
And on the other pieces no growth is visible. But I think I have at least one more piece left, that looks very promising.