The conjecture on infinitely many nontrivially squared consecutive prime pairs

Let $p$ and $q$ be consecutive primes with $p>q$. Two primes are nontrivially squared if there exist natural numbers $(x,y)\ne(0,0)$ such that $p^x-q^y$ is a perfect square.

Nontrivially squared prime-pair conjecture. There are infinitely many consecutive primes $p$ and $q$ ($p>q$) which are nontrivially squared.

The source motivates this using primes of the form $n^2+1$ and notes that Landau's conjectures would imply both this conjecture and the preceding one. Its status is not established in the source.