For those who’s having the same issue: after investigation and help from Asana side and especially @Ross_Grambo it turned out that issue is with Asana internal libraries that don’t fully support HTTP 2.0 protocol.
While this issue is being fixed on Asana side a workaround is to downgrade to HTTP 1.1.
On my shared hosting it was enough to add next lines to `.htaccess. file:
Header edit Upgrade (.*)h2,h2c(.*) "$1$2"
Header edit vary (.*) "$1 do"
Thank you @Ross_Grambo for all the testing and help!