NextJS Cognito authentication with SST

Question

I'm building a NextJS application and deploying it to AWS with SST. I understand that in this context no API Gateway is generated, but instead a URL is linked directly to the server lambda. There's still an authorizer configuration available in this case. I would like to have an authorizer using AWS Cognito that will redirect me to the home page if I access a path starting with /auth without any valid credentials, but so far I haven't been able to even configure the relationship between Cognito and my NextJS app.

Here's my SST stack so far:

import { SSTConfig } from 'sst';
import { NextjsSite, Cognito } from 'sst/constructs';

export default {
  config(_input) {
    return {
      name: 'myapp',
      region: 'us-east-1',
    };
  },
  stacks(app) {
    app.stack(function Site({ stack }) {
      const site = new NextjsSite(stack, 'site', {
        runtime: 'nodejs20.x',
        cdk: {
          server: {
            logRetention: 30,
          },
        },
      });

      stack.addOutputs({
        SiteUrl: site.url,
      });

      new Cognito(stack, 'auth', {
        login: ['email'],
        cdk: {
          userPool: {
            standardAttributes: {
              givenName: { required: true, mutable: true },
              familyName: { required: true, mutable: true },
            },
          },
        },
      }).attachPermissionsForAuthUsers(stack, [site]);
    });
  },
} satisfies SSTConfig;

At this point .attachPermissionsForAuthUsers(stack, [site]); definitely is not doing what I want it to do (as I cannot bind it to any specific path) and is giving me Error: The specified permissions are not supported.

I cannot find any docs that would show how to do it. I'm starting to wonder if it's possible at all or if I should just integrate cognito directly into the code of my NextJS app...