No, I don’t think the White House website would be incorrect.
I think the confusion is that there is no photo evidence that he signed it and they stated in the Rose Garden that he would be signing it later.
Well, IMO, if the WH posts on their web page that POTUS signed it, there's an end to it. Let doubters prove otherwise. (Not to be too contentious. 😉 Speaking for myself. I suppose the Russians could have hacked the WH web site.)