I made up a moderator for the tube. I have been cutting down to size a 2m x 1m piece of 10mm thick HDPE sheet. (cuts easy with a jig-saw, at that thickness).
There does appear to be a higher count. It's a bit 'bursty', but as per prior discussions on fusor.net, background neutron counts with moderator are generally muon stuff generated in the moderator, rather than 'fast neutrons' being moderated, because it's mostly thermal neutrons at sea level - presumably on account of all the water in the air.
I'd say this morning it went from around 4-6 counts per min (2~3 per min after my aggressive thresholding), to around 7-12 (4~6 with thresholding). Watching the 'scope, I got the impression it wan't quite 'double', as those figures suggest, but that was the actual count from 5 mins of each came up with. On 'rate' I'd have estimated a 50% increase.
I'll do some long runs once the circuit is made up.
My original plan was to have 4 layers with an aim of 4cm worth of moderator, which I might move towards, but there again if I point the 'diagonal' towards a neutron source, then any neutron reaching the tube would've had to go through 4 cm worth of moderator for most paths to there. So, for now, I think I'll see how this goes and chop up more HDPE only if needed/feel like experimenting.