Preview with Pipelines/Releases shows 401 error

I get a 401 error in the console when I open the preview of a page for the “Live” preview environment.
I assume the problem is that the preview wants to use the public token for the draft version.

I set up the preview environment “Live” that points to the live URL and a stage “Live” next to the default “Preview”.
I assigned API keys to the Live stage – both a public and a preview key. The preview stage has a preview key.

The preview for the “Live” URL always tries to make a request with the “public (Live)” key and always gets a 401 error.
The request goes to<public (Live) token>

The preview of the default “preview” URL works.
But that goes against my localhost.
I want stakeholders to see the live page in the preview screen.

What am I doing wrong?
Can I control which token the preview URL uses?

Technical context:
I’m using Nuxt.js as static site. The preview first shows the correct page for a split second (the pre-rendered static files) and then tries to make the Storyblok calls to preview draft content – that results in a 401 and the page shows an error.