What’s a lightweight way to synchronize CLIPBOARD -> PRIMARY selections?

My use-case is that I want that whenever I copy something to CLIPBOARD, it is also saved in PRIMARY. It’s mostly assumed that to copy something you need to select it so most of the time this is not needed.

However, sometimes I just click the classic “copy to clipboard” button and get something to CLIPBOARD that it’s not in PRIMARY. I use Shift+Insert a lot for pasting and having to track which selection I’m using makes me confused.

I know there are a tools like clipit or parcellite that do something like this, but I wan’t something without a GUI, something like a simple systemd service I can launch and forget.

I tried using a systemd service for autocutsel configured like

ExecStartPre=autocutsel -f
ExecStart=autocutsel -f --selection PRIMARY

However this also synchronizes PRIMARY -> CLIPBOARD, which breaks some very usual workflow like selecting text and then replacing it with the contents of the clipboard.

I’ve looked for this option in the manpage of autocutsel, but I find it kinda confusing, with a lot of mentions to cutbuffer (which I don’t think it’s used anymore) and Windows which I don’t use. So I don’t even know if this is possible with autocutsel.