People mentioned exchanging 1 text with a T3 would raise invitation, but I’ve had a couple of phone calls with a T3 and never experienced any such increase. I also have roleplayed in the same room as one for up to 2 hours, but no increase. I feel that if your character roleplays with them regularly and has built a more personal relationship with them, that may be what boosts the rate of increase to be impossibly quick.
Another theory that came to mind is if it might not be that your invitation goes up by a compounding of interactions with a T3+, but instead if your character has fresh history with or a close/frequented relationship with one, then your base invitation level can just never go below a certain #.
To help counteract the rate of invitation gain, my character specifically tries to help others lower invitation level through RP or public events geared towards that. I’m just not sure how effective these active goals are versus him simply being a T1 in a room of T2s.