Didn't notice for a while since the HLK Client PC is running headless which wasted quite a lot of time. The HLK Client PC is at another desk in the lab. Both the HLK controller running Windows 2019 Server Std. and the HLK Client PC running Windows 10 run headless. The operator runs the tests from a work-place PC with access to the same network as the HLK machines.
None of the suggestions from copilot (pinging the HLK Client PC or HLK controller IP address, restarting services, reinstalling, resetting etc.) worked. Deleting and recreating the project in HLK studio seemed to resolve the problem for a while but this was probably just coincidental. The real solution is in the next paragraph.
In our case the problem was caused by the HLK user (DTMLLUAdminUser on our machine) not auto-logging on to the HLK Client PC after the HLK Client PC was rebooted remotely from WinDbg running on the work-place PC. Analyzing test failures and installing corrected driver versions was all done from the work-place PC. Multiple (random number of) HLK Client PC reboots from the debugger or simply attaching a monitor and clicking on the sign-in button on the HLK Client PC whenever auto login didn't succeed solved the issue reliably.