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I am amazed!
Only had one mistake in there, one diode stack was oriented the wrong way, blocking any current flow. Luckily it was only three stages down from the top, so rotating it was fairly quick.
To test the thing I supplied 1.5 kV through 2 MOhm to the output, polarized so that the diodes are in forward direction.
This plot shows the measured stage voltages resulting from diode drop.
Looks pretty damn linear to me!