2 CalDigit TS3 Plus docks, 3 external monitors

MBPro, 2019 with 4 Thunderbolt 3 ports. Big Sur 11.3

To support two external monitors I have been using a CalDigit TS3 Plus dock for a year with zero issues.

I want to add a third external monitor. I have found references that this can be done by using a second TS3 Plus dock connected to the other Thunderbolt bus on the opposite side of the laptop. That sounds reasonable, but it is not a plug and play scenario. I can get left and middle displays to work if I connect just the left dock, or I can get the middle and right displays to work if I connect just the right dock. (This is a bit odd since the left dock has the two monitors connected to it, but the right dock has only the right monitor.)

It seems it might be possible.

Any advice?