Not sure on the 737s, but for the Airbus there is an element of truth to this rumour.
BA recently retired the (HF equipped) 757 fleet. This leaves just 5 LHR based shorthaul aircraft (Airbus A320) with HF fitted and three of these are to be retired by the end of the summer.
The HF fit has been very useful during both French & Spanish ATC industrial action. They allow routes to be flown through oceanic airspace avoiding the affected areas.
I know that a business case was being put together to get additional HF capacity, but had not heard any outcome or time scale.