Trump's press secretary just admitted or at least inferred that he has no "proof" of his version of the phone call. 

My take is that, being a paranoid type, he probably does tape his conversations but belatedly has realized that it's now a double-edged sword. After claiming he had "tapes" of his conversations with Comey, he then said he really didn't - likely because they proved Comey's presentation of events. Now, even if he is right about how events transpired in his phone call to the parents of a dead soldier, he can't produce tapes because then his denial about the Comey tapes will be exposed as yet another spectacular lie. 

When one lives their life by lying as Trump does reflexively, eventually one finds themselves floundering in the deep quicksand of bs one has created.