The property TrimTrailingSpaces works during editing, but when the file is saved to disk all lines are trimmed in
TAdvMemoStrings.SaveToFile:
...
BeginUpdate;
for i := 0 to Count - 1 do
begin
tr := TrimRight(Strings[i]);
if Strings[i] <> tr then
Strings[i] := tr;
end;
EndUpdate;
…
This could be a feature request or a bug report. I imagine since it's been this way for likely a long time, I opted for a feature request.