I put on strip tesa on the sensor. Now it works great
Thank you for the advice 
I gave it a try for 6 weeks - with tesa/ahesive film, the fingerprint works reliably in all conditions, i.e. usually 1st try is a success, while after removing the film, around 50% of login tries fail completely, i.e. using several fingers several times until fingerprint is de-activated due to too many failed tries. Moreover, with film, I virtually never saw the message “sensor dirty” while in fact, it was clean - without film, this happened many times a day.