Andrica's conjecture for consecutive prime powers

Let $q_1,q_2,\ldots$ be the consecutive prime powers. Andrica's conjecture for prime powers. The inequality

$$\sqrt{q_{n+1}}-\sqrt{q_n}<1$$

holds for every $n$. This is a denser-sequence analogue of Andrica's conjecture for consecutive primes and remains open in the stated generality.