I forced myself to give this a break for a few days - but now it's time for a new theory....
It might be Alleyn Park in West Dulwich - down the bottom end where it meets the Paxton Green junction. It's not Herne Hill, but bear with me.
The road was laid out at the right time. (1850s - I checked my suspicions on the house being no later than 1870 by using the dating tool on
www.bricksandbrass.co.uk) More about the building of the properties along the road here:
South Croxted Road and Alleyn Park
Nearly all of the original houses are gone but most were still standing in 1944.
We can't be sure what they all looked like, but Pissarro painted a view of some of them across the fields in 1871. They begin to look quite promising.
Then there are the specifics of the design - which is very particular. Three stories - with twinned round-headed windows over 4-pane sashes over a hexagonal bay in the gabled wing. I've been using my daily government-sanctioned run to do a bit of exploring and nearly every house of this age and type just has one central window in the second floor gable end.
Now it gets interesting. No 38 is still there - built into Dulwich Prep School. Some of the details are a little different but the form and proportions are right. Don't be fooled by the left hand wing - it's a modern extension, as can be seen on Google Maps.
What's even more interesting is that the back of the house shows the same arrangement of windows and bay. So we could be standing behind these houses after all.
Now take a look at Numbers 3 and 4 Alleyn Park on the 1944 map. They have the same plan as No. 38. They're far enough apart. They're a mirror pair. They have long back gardens.
I'm not saying these are the houses for sure, but are we getting warmer?
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