Breaks first Commerce "Add to cart" for anonymous users

[Fonant]

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.