I've heard of instances where Branch employees will directly offer you Emerald LTF if your company is on ICICI Corporate tie up list,For Corporate your company needs to have tie-ups with ICICI and your Corporate RM will help you to get the card LTF no other channels work as per my RM.
In the case of Govt. Organizations it is somewhat easy and flexible to get without meeting the full criteria there is an option "Special Categories" where they get approval from RH. At the time of application, they select a paid variant card but on approval, it is converted to LTF instantly as they take the approval beforehand. These works till Sapphiro LTF. In the case of Emerald not verified.
Alternatively I've also heard of people who have corporate tie up(atleast they think so) but couldn't get hold of corporate RM or couldn't get Emerald LTF at all