The conjecture on infinitely many trivially squared consecutive prime pairs

Let $p$ and $q$ be consecutive primes with $p>q$. Two primes are trivially squared if $p^x-q^y$ being a perfect square implies $(x,y)=(0,0)$.

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

This extends the observation that infinitely many twin primes $p,p-2$ with $p\equiv 3\pmod{4}$ would provide such examples. Its status is not established in the source.