There is a conflict between the living-pterosaurs movement and the standard model of ancient universal extinction of pterosaurs. The idea that no pterosaur fossil has been found above the Mesozoic strata cannot be used as if evidence against living pterosaurs. Why not? Those who have been searching for living pterosaurs do not, for the most part, believe in the axioms that are at the foundation of the concept of such strata. On the other hand, it seems that those who so use the concept (of the Mesozoic strata), to dismiss the possibility of living pterosaurs, do so without taking into account those axioms.
When a pterosaur fossil is found, it’s normally assumed that the creature lived in the Mesozoic time period; the layer in which the fossil was found is then classified as Mesozoic. If some pterosaurs were living in modern times, and one became fossilized (let's say a few centuries ago), its discovery would lead to it being incorrectly dated in the Mesozoic; the strata would then be assessed or re-assessed as “Mesozoic.” That's why standard classification of strata cannot be used as if evidence against the idea of living pterosaurs: It would be circular reasoning.