How do we deploy a v3 studio for a static site?

[dwkns]

I'm using 11ty to build a static site.
I'd like to host my Sanity Studio on mysite.com/studio.

I've set up sanity build to output to the correct folder, but when you visit mysite.com/studio there are 404's and the studio doesn't load.

Looking in the generated HTML file all the links are relative to the site root eg ="/static/manifest.webmanifest", not the containing folder where I want them ="/studio/static/manifest.webmanifest".

export default defineConfig({
  ...
  basePath: '/studio',
  ...
  })

Doesn't appear to have any effect.

If I manually change the links to be relative in the generated HTML file to ="./static/manifest.webmanifest" or even ="/studio/static/manifest.webmanifest" and then visit mysite.com/studio the studio loads but I get an Tool not found: studio error.

I can click on Desk and then use the studio but it appears that the studio thinks it should be at the root URL /

Question 1
Can I specify to the sanity build command I want relative, not absolute links? If so how?

Question 2
How do I tell the Studio it should be at /studio/ not at /

OR
If I'm going about this the wrong way, how should I be including my Studio in a static build?