Ok guess I've solved the issue! At least, found a temporary solution until they will fix it.
What I did - I've found pirate torrent for 1.5.6, downloaded only steam_api64.dll file (there's no need to download all 40gb torrents, only this 600kb file) - renamed the original file to another name (just in case), copied the pirate steam_api64.dll file to the game folder (for example, C:\Program Files\Epic Games\Chickadee\bin\Win64_Shipping_Client ) and the game started!
I'm not sure if it's eligible to share the pirate .dll files here (I'm pretty sure that it's not) but I've placed it here for 7 days: