Cookies not set when Breeze is active

I’m writing a simple plugin for login with an external service and it does not work when Breeze is on (active). I narrowed it down to a simple problem:
This code
`setcookie(“TestCookie”, ‘Hello’, time()+3600);
— When Breeze is OFF the cookie IS set
— When Breeze is ON the cookie IS NOT set

I used the Chrome Dev Tools and noticed that:
— When Breeze is ON the WP does not send the Cookie header at all.
Please note that the URLs are added to the Breeze setting in the “Never Cache these URLs” field. No difference at all.

One noticeable exception when it kind of works:
— If for some reason there is at least one WP cookie already set, such as “wordpress_test_cookie” it will work because the WP will start setting up the cookies properly, i.e. will start sending the cookie headers including mine.

How to solve the issue, any suggestions?