How to reload localStorage saved aws-amplify auth?

I am successfully using AWS Amplify and AWS Cognito for my spa authorization. The only issue is that when I refresh the page, the auth seems to be lost, even if I see amplify and cognito items in LocalStorage.

This is my configuration section:

Amplify.configure({
  Auth: {
    oauth: {
      domain: 'xxxxx.auth.eu-west-1.amazoncognito.com',
      scope: ['phone', 'email', 'profile', 'openid'],
      responseType: 'code',
      oauth.redirectSignIn: 'http://localhost:3000/',
      oauth.redirectSignOut: 'http://localhost:3000/',
    },
    region: process.env.REACT_APP_REGION,
    userPoolId: process.env.REACT_APP_USER_POOL_ID,
    userPoolWebClientId: process.env.REACT_APP_USER_POOL_WEB_CLIENT_ID,
    storage: localStorage,
  },
});

In practice, I don't know how to ask amplify to use localStorage when user is authenticated and app restarts (page reload).
Possibly I should use Auth.currentAuthenticatedUser(), but I suppose I can't call it instead of Auth.configure()...