Try specifying the access token with x-access-token
when pushing GitHub pages
#2807
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refs #2802 (comment)
I don't have a means of testing this, it's just a guess based on some of the Github actions for pushing pages, e.g.
https://github.com/peaceiris/actions-gh-pages/blob/0b7567fde6f7517edcc13d8ffa2d89cd8734d47c/src/set-tokens.ts#L100
https://github.com/EdricChan03/action-build-deploy-ghpages/blob/3183472d918d9ea82237bf4f2c416e5c8c13a394/entrypoint.sh#L32