Since TestStand is an ActiveX Server and i am not aware of any well written ActiveX wrappers for Linux, i doubt that it will work.
Officially, TestStand is only supported for Windows. So if you achieve by any what-so-ever-open-source-code to install and run TestStand on Linux, you wont get any support from NI if anything unexpected happens.
Has the advent of .NET Core on Linux changed anything in this respect?
Like, re-implementing TestStand based on .NET Core or using a COM infrastructure which may possibly lie underneath .NET Core on Linux (although this https://docs.microsoft.com/en-us/dotnet/standard/native-interop/cominterop points into the direction that COM is only supported in .NET Core on Windows).