I am honestly not the person to know the answer to this--I expect only an insider could say for sure. I have seen in the past that upgrades are available at check-in via the kiosk or by asking either at the check-in area or even at the gate. I have asked about the latter a few times and it was always more money than I wanted to spend.

My recent experience is that it did not show up at OLCI or at the kiosk. When I asked the assistance agent he was able to pull it up on his screen and tell me both how much and take payment. However, there was a glitch and support had to be called and some sort of work-around needed to be implemented, which is why I wonder if there was a general glitch in the process. I know there were a couple of other people who boarded very late who had literally just upgraded at the gate, so it was possible even then. The PE cabin was not full and I wonder if that was related to the same issue.

My conclusion is that it is always worth asking, and if a seat is available they will find a price for it. Pre-pandemic, I think the business cabin tended to book full with paid passengers as it is so small, so the current times may be a little different. I have done gate-side upgrades on LH from PE to business a few times and it was always the same prices--so, a set price--and easy to do so long as there was space available.