LSWS does not treat HTTP/1.0 or 1.1 differently on this matter. Have you changed any configuration?
What testing tool are you using?
You should redirect the result to a file, as the last line of LiteSpeed output is too short, just "missing", without "\n", so it was overwritten by the command line prompt. It is there, just not visible from command line.
I think the issue you experienced is browser related. Have you tried it with other browsers?