Infinitely many imaginary quadratic fields with class number divisible by a prescribed odd integer

Let $p$ and $q$ be distinct odd primes. Let $n$ be a positive odd integer, and let $m$ be a positive integer that is not an $n$-th root of any rational integer. Consider the imaginary quadratic fields of the form

$$\mathbb{Q}(\sqrt{p^2-mq^n}).$$

The authors' conjecture. For each such $p$, $q$, $n$, and $m$, there are infinitely many imaginary quadratic fields of this form whose class number is divisible by $n$. This extends the numerical phenomenon established for the paper's explicit family, while the general assertion remains open.