I installed the latest stable release of ConEmu. I opened a Cygwin shell and used SSH to connect to a remote computer. I ran Byobu on the remote computer. I noticed that its status bar would shift upward, overwriting the entire screen after some time.
This problem is documented here. The solution suggested by the ConEmu creator is to use the ConEmu/msys terminal connector plugin. Using this connector is however a bit complicated. But, according to this post, it has been rolled into ConEmu starting with the 170705 release. I checked and found that the stable release I had installed was 161206. The stable release was years behind!
Seeing no recent stable version available, I opted to download a preview release from here. This release was dated later than 170705. byobu worked fine through SSH from Cygwin after that!