Not that I work at Icici or taking their side
but generally Debit Card LTF are always tied to the respective Program / Account unlike CC LTF's.
A Private Banking DC is LTF as long as you are a Private Banking Customer. If the account is downgraded than the variant of Debit Card for that particular Account will be LTF. Charges can be applied for higher variant DC.
It is the same for Hdfc, Axis (don't now about Kotak). In fact, Hdfc will charge you on a Pro-Rata basis >> meaning if you have imperia debit card and your acc is downgraded to a non-premier banking account, then for the number of days remaining in that year wherein the card is issued, will be charged a fee.
For eg - 365 days in one year. Imperia downgrade after 100 days. You will be charged fee for remaining 265 days.
On the flip side/ a positive note
Icici charges I think only ₹200 per annum (not fully sure on this), so Ig everyone can just keep the Pvt Banking DC.