The subdomain cannot be set for the cookie

Environment

Operating System: Darwin
Node Version: v20.15.0
Nuxt Version: 3.12.4
CLI Version: 3.12.0
Nitro Version: 2.9.7
Package Manager: yarn@1.22.22
Builder: -
User Config: css, router, debug, head, plugins, modules, vuetify, i18n, typescript, sentry, auth, pwa, facebook, gtm, runtimeConfig, dayjs, build, hooks, vite, eslint, proxy, compatibilityDate
Runtime Modules: @nuxtjs/i18n@8.4.0, @kevinmarrec/nuxt-pwa@0.17.0, @sentry/nuxt/module@8.22.0, nuxt3-meta-pixel@1.0.9, @nuxt/image@1.7.0, dayjs-nuxt@2.1.9, @nuxt/eslint@0.5.0, @zadigetvoltaire/nuxt-gtm@0.0.13, vue-clipboard3@2.0.0, @pinia/nuxt@0.5.3, @nuxt-alt/auth@3.1.7, @nuxt-alt/proxy@2.5.8, @nuxt-alt/http@1.7.10, vuetify-nuxt-module@0.18.2
Build Modules: -

Nuxt Config

auth: {
    strategies: {
      auth0: {
        domain: process.env.AUTH0_DOMAIN || 'xxxxxxxx',
        clientId: process.env.AUTH0_CLIENT_ID || 'xxxxxxx,
        audience: process.env.AUTH0_CLIENT_ID || 'xxxxxx',
        scope: ['openid', 'profile', 'email', 'offline_access'],
        responseType: 'code',
        accessType: 'offline',
        grantType: 'authorization_code',
        cookie: {
          name: 'auth._token.auth0',
          options: {
            secure: process.env.NODE_ENV === 'production',
            sameSite: 'lax',
            path: '/',
            domain: '.quickk.local',
          }
        }
      },
    },
    redirect: {
      login: false, 
      logout: '/', /
      callback: '/authentication/callback',
      home: '/',
    },
  },

Reproduction

When setting the domain for the cookie with the option domain: '.quick.local', it doesn't work.

Describe the bug

When I use domain: '.quick.local' to store cookies for subdomains, the cookie is only stored on the main domain quick.local. Subdomains like test.quick.local and admin.quick.localdo not work.

cookie info in main domain

cookie info in sub domain

Additional context

No response

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