He needs to show proof, and he doesn't seem to have done so
He actually doesn’t. He presents his argument (and you’re right in that the more proof he has the stronger his argument) and the other side counters the argument. The judge decides which argument is more likely than not.
Given MGN have previously admitted unlawful activity (including in this actual case), paid settlements to others and have hundreds of similar claims made against them, they’re starting from a considerably worse position than PH and the others.