When you log in with an AJAX request, the full page cache is not updated when using Amasty_Fpc. This happens only on the page on which user log in.
The problem is observed with many modules (for example Youama_Ajaxlogin).
Excluding blocks from the cache as described in the documentation gives no results. All options from the documentation were tried. Hole-punch does not work in this case also for standard blocks like "welcome" "top.links".
How update full page cache when log in with AJAX request when using Amasty_Fpc?