One thought on this - if they're looking for it they'll find plenty of evidence that the people aren't from our world - in any electronic equipment they're carrying. We know that the L&C world has big differences in its major industrial players, the really major one being Lexcorp with a finger in all sorts of pies.
You could maybe fake up a computer complete with operating system etc. which will probably be a bit different from those of our world, but inside it there will be things like chips that have been fabricated by LexCorp and have their copyright data on the masks, only readable with an electron microscope, hard disks of slightly different design to anything made in our world, etc. And tooling up for a fake like that and ending up with hardware that actually worked would require a global effort by hundreds of manufacturers. It's easier to believe that they come from another world.
