Hi Pavel,
I don't know why the problem would be browser specific other than perhaps cookies being different, but this issue was indeed a change in behavior after the recent ASP.NET security update. We patched the 2.3.7.6 release of mojoPortal to solve this problem as discussed on this thread where it was first reported. Clearing the browser cookies might also solve it in some cases.
There is also another issue from this security update that can cause an error in sites that have a lot of roles as discussed on this thread.
Hope that helps,
Joe