Conjecture on class numbers in the family Q(sqrt(p^{2r}+1))

Fix a prime $p>2$, and for each $r\in\mathbb{Z}^+$ consider the real quadratic field

$$K_r=\mathbb{Q}(\sqrt{p^{2r}+1}).$$

Class-number infinitude conjecture. There are infinitely many $r\in\mathbb{Z}^+$ such that $p$ does not divide the class number of $K_r$.

The conjecture is motivated by computational data and would imply an infinite family of fields in which the paper establishes Greenberg's conjecture under its non-Wieferich hypothesis.