Below is a link to another similar external coders blog site, where this sort of thing has been discussed.
https://stackoverflow.com/questions/59983825/launch-microsoft-edge-in-private-incognito-mode-in-net
Going by examples at that reference, I created the following code within an iLogic rule. When I ran it, it seemed to work OK, but I believe that the 'InPrivate' mode is disabled where I work, due to corporate IT Dept. restrictions, so I couldn't fully test it properly. I almost never use MS Edge, and use DuckDuckGo Browser instead. It can even be installed & used without 'administrator's' rights, at least for me. I have been a fan of theirs since they started, back about 15 years ago, as an alternative to g@*gle .
Dim oProcStInfo As New ProcessStartInfo()
oProcStInfo.UseShellExecute = True
oProcStInfo.FileName = "msedge.exe"
oProcStInfo.Arguments = "-inprivate https://www.autodesk.com/"
Using oMSE_Proc As Process = Process.Start(oProcStInfo)
Logger.Info("ProcessName = " & oMSE_Proc.ProcessName & vbCrLf & _
"Process ID = " & oMSE_Proc.Id.ToString & vbCrLf & _
"Process MainWindowTitle = " & oMSE_Proc.MainWindowTitle & vbCrLf & _
"Process SessionId = " & oMSE_Proc.SessionId.ToString)
' oMSE_Proc.Close()
End Using
Wesley Crihfield

(Not an Autodesk Employee)