According to the "Discover BT Halo" found by clicking the button in the menu bar at the top of this page, the price promise is
"Our price promise
Your bill won’t go up when your contract ends, and you’ll never pay more than a new customer."
Yes, never more than a new customer would for a halo package...….
This is for Halo plans only, so the price during the contract will not increase when the contract ends, It has nothing to do with what new customers pay, It's the price you agree to have Halo at when you renew the contract.
Other services such as BT Sport, BT TV or BT Mobile are not connected to the Halo plans, so any discount agreed on them will stop after the agreed term, you'll need to renegotiate.