On a Conjecture Involving Twin Practical Numbers

In this paper, we state a conjecture that for any positive integer $p$, there are infinitely many practical numbers $q$ such that both $q^p$ and $q^p + 2$ are practical numbers. We prove the conjecture when $p$ is a prime number between $7$ and $100$, refining a result involving twin practical numbers of Wang and Sun. We also prove several results related to this conjecture.