HLS Livestream without relative time on-timeline-hover

[SpraxDev]

Hi,
for my project I am taking a video file that browsers do not support and turn it into a HLS Live Stream.

So you can already start watching the video while it is still being processed in the background - That means that per real live second the stream can end up producing 28 seconds of content.

Because this is not a live stream in the common sense and trying to mimic a normal video player I'd like the timeline to not show -12:20 but 2:40 like for a normal video. (old video segments are not deleted so Player#currentTime seems to always return the correct seconds)

Is there an option or maybe a way to hook into that behavior?

Thanks in advance!