OK, now I have the answer.
Today I had to stop the long-term test, so I did the measurement at VCM = 10 V. So far, with 10 FFT averages, it looks pretty much the same as with VCM = 0 V. So the extra noise comes from the ADC, and not from the resistors. If they contribute something, it's really negligible.
The voltages across the 20 K elements (R1-R3-R5-R7) and 3.5 K ones (R2-R4-R6-R8) in the R array has the following dependence on VCM (input):
As you can see, due to VOCM=2.5 V, all 8 elements see zero voltage when VCM (input) = 2.5 V. At VCM=+10V they see 3 times higher voltages than at 0V.
And here's what the noise spectra look like. For completeness, the spectrum with internal ADC short is added.
(Edit: replaced the 10V CM plot line (red) with one averaged over longer time)
There's no need to take further measurements to confirm it. Those Vishay resistors definitely have noise index of -50 dB or lower.