The root key here is Gm, but the other chords bring in other notes and scales and - as ever - I don't have the theoretical basis to be clear what would be "right" here. I little improvisation over it shows that keeping in Gm pentatonic sounds about right, but a bit dull. The other approach is to work out what notes are in the chords and work with those (in fact, B, whose knowledge of music theory is better than mine, suggested this as well). I probably should have done this to advance my knowledge and understanding, but I didn't have the time and couldn't really be bothered.
I experimented for a while with variations of the chords at different positions to see if the transitions inspired anything, but couldn't quite make it work to my satisfaction (one of the other participants did the same and ended up with a gorgeous sounding spy theme, much better than I would have been able to do). However, during this, I hit on the idea of playing with octaves, and decided to make this the basis of my solo.
I played through the track multiple times and eventually the shape of the solo worked itself out. So I suppose we could consider it "composed" rather than improvised. I recorded it in sections so I wouldn't forget it, and then played along with myself to get the whole solo in my head. I liked the sound of this, so I decided to have two guitars at points. The important thing there is that the two guitars are two different takes, not just the same take duplicated, because the tiny differences in timings and intonations is what gives it a sense of space.
The final take is a single pass through, since I'd learned the whole thing, and the second guitar is used to "pop" some of the key notes, as well as providing a harmony in the last four bars. Both guitars were recorded dry, just through the amp. I added a bit of EQ and compression in Reaper, but no reverb since it sounds nice without it. I used a touch of reverb to the harmony guitar to give it a slightly different position in the track. Finally, when it got to the harmonies in the last bars, I panned the guitars slightly left and right, to emphasise the change.
This was a bit of a struggle, mainly because I couldn't hear anything over the track for a long time. But the end result, a product of a solid three hours right at the end of the time allowed, is something I'm rather pleased with. It's understated and works with the track rather than sitting on top of it, I think.