there can be no "proof" about such things. Only more or less plausible evidence, with gaps, and it is not dogma or needed to support anybody's faith. But, if it was the cross of Spartacus found with the same level of evidence, nobody would make any fuss, and just accept it as the most plausible candidate from a plausible place/time and be happy with it. So your sources from skeptics tell there was a titulus, but separated from the cross.

And probably the pieces of the crosses were not fixed together (and most sources tell the titulus was not fixed)- many people think the horiz. bar had a wedge to just be put over the vertical pole and just remain there by gravity. So the cross was indeed a T, and only the titulus was above His head. One reason: a full cross is too heavy to be carried by one man.