Recaptcha not showing up when making use of PrivateTempStore

When making use of PrivateTempStore, the reCaptcha does not show up, e.g. when used in a webform.

If I execute the code below in my custom module, e.g. to save something for my user:
\Drupal::service('tempstore.private')->get('mymodule')->set('mymodule_mykey', 'myvalue');

Then the captcha does not show up in my form anymore. Working with cookies is a workaround but I think it was worth to mention this issue here.