Breaks first Commerce "Add to cart" for anonymous users

For some reason clicking "Add to cart" for a Commerce product the first time does not result in a new session cookie being saved. Although the product is added, there is no visible sign of this, confusing the customer.

I think the Product page is being cached for the "no cookie" state, and not refreshed when Commerce sets a cookie for the cart.

Turning off lscache in the Drupal admin settings fixes the problem.

Start by reproducing the first anonymous Commerce "Add to cart" with LSCache enabled, then compare the session-cookie and cached-page behavior with LSCache disabled. Done means the first add visibly establishes the cart session and remains correct without disabling LSCache; the issue names no files or tests.

Written by the indexing model from the issue text.