I've given part of it a quick wire brushing, but before I go any further thought I would try out some supposed rust
remover (rather than converter). I'm sceptical of any method other than physical removal of rust, but thought I'd give some a try as it's always difficult to remove the rust at the bottom of pits.

The idea is you put it on as a thick layer of gel, and then leave for a day or so.

Given that I had removed any flaking rust, I cleaned off the gel after leaving it overnight. It's a poor photo, but I can see no trace of rust underneath where it was. The black areas are paint that I have not yet removed.
I'm actually quite (reluctantly!) impressed, so will probably be using it on the rest of the screen surround once I've got rid of the flaky stuff. Will try and get some more photos, of a better quality