BMW E28 speedometer problems

Hello, 5 Series BMW E28 Tacho problem: – Speed indicator defective – Speed indicator defective – Average consumption indicator defective – Service lamps light up (with the reset tool they can be deleted and after that only the inspection lamp lights up, but everything else works permanently. The problem suddenly occurred. I read that sometimes the battery on the motherboard runs out and it can come to problems. Questions: 1) Someone has had this already with the E28 with the Batteri e on the mainboard. Can that be the reason? 2) If I buy another / used tacho, does this work “plug and play” with such an old car? Or is it the process? 3) Could someone find out for me whether the tacho was installed in this car ex factory? Before 10j it was for repair, I could take the documents. Then send chassis number and tacho number via PN. Would be great if someone has useful tips or can help. Please don’t go to BMW, or Keep your hands off of it” ;p. Thx