I received PCBs yesterday and this afternoon I had a partially working board. I decided not to populate the LTC6993-1, the DFF and the 3.3 LDO. I was only interested in determining which band-aid fixed the problem in the simplest manner, and for that all I needed was a functional LTC4041. Refer to the schematic in the previous log for component references. A Raspberry Pi Zero W was used as a load in all of the testing.

R18, R19, and C12 were populated since I knew the impact of not having them in place from previous tests. R21 was a short and I left C13 unpopulated to see if it was needed or not. The value of C2 is 22nF, which decreased the TMIN-BACKUP time that the LTC4041 spends ignoring the power-fail comparator after it has tripped.

It turns out that this was all that was needed to fix the problem. I suspected that the relaxation oscillation was caused by the 500ms TMIN-BACKUP time being too long, which allowed the output voltage of the UPS to decay to zero during that interval. Decreasing TMIN-BACKUP to less than 50ms doesn't allow the output voltage to decay so far and can reach a point where it is above the UVLO threshold of the LTC4041 after one or two cycles. Here's a trace of input(CH2) and output(CH1) voltage during power-up.

You can see that the output voltage jumps a couple of volts before the PFI comparator switches off and forces the LTC4041 into backup mode -- but the voltage on the supercap is not high enough and the output voltage is not high enough either to allow the boost converter to run. So the output voltage falls back a bit until the TMIN-BACKUP time ends and the input switch is closed again -- this time the output voltage has climbed high enough to keep the booster enabled (or the PFI comparator doesn't trip because the input-output differential is not enough to drag the input voltage down below the PFI threshold.) In either case the output voltage is sustained and the UPS settles into normal mode with the input switch closed and the boost converter off.

I then populated C13 with a 220µF/6.3V tantalum, but it did not appear to make any difference in operation. I did not investigate the options with C14 and R21 -- I will remove those going forward. I've got enough confidence now to finalize the UPS design.