[California] 2-year limitation very close for accident I was at fault for. Why no demand letter?

We are only days away from the 2-year limit. My insurance says there hasn't been a demand, and I haven't been served. Are they waiting until the last second to serve me or something? I feel paranoid waiting to be served, so I'm anxious for that to be over.

I have $250k bodily injury, which should be sufficient for what happened, so I'm just confused that it hasn't been resolved. Any idea what's going on in the background that I don't know about.