Rewrite not working on deployed app

[cobalt python]

I have this rewrite config:

/** @type {import('next').NextConfig} */
const nextConfig = {
  async rewrites() {
    return [
      {
        source: "/",
        destination: "https://wax-wanderer-7334.typedream.app/",
      },
      {
        source: "/:path*",
        destination: "https://wax-wanderer-7334.typedream.app/:path*",
      },
    ];
  },
};

module.exports = nextConfig;

Locally everything works, app loads. I tried both pnpm build & pnpm start as well as pnpm dev. But when I deploy to Vercel, it doesn't work.

Here is original page:
https://wax-wanderer-7334.typedream.app/

Here is page with desired rewrite:
https://www.bubbleteabrno.cz/

According to DevTools other stuff returns 404, but... why if it works locally?

[earnest stirrup]

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[cobalt python]

As I wasn't able to fix this I went with the iframe solution, but it would be better if this would work. Any ideas?