I observed the following result empirically based on numerical evidence.

**Conjecture**: Let $[x]$ denote the greatest integer function. If $a_n$ is an strictly increasing sequence of positive real numbers such that $a_1 \ge 1$ and $\lim_{n \to \infty}\frac{1}{a_n}=0$ and $\lim_{n \to \infty}\frac{a_{[nt]}}{a_n} = t$, for every real $0 \le t < 1$. I observe empirically that
$$
\lim_{n \to \infty} \frac{\ln\Big(\frac{\sqrt[n]{a_n}}{\sqrt[n+1]{a_{n+1}}}-1\Big)}{\ln n} = -2.
$$

**Motivation**: The sequence of natural numbers $n$, prime numbers $p_n$ etc. I am particularly interested in the limit involving the sequence of primes $p_n$. This is a generalization of the Firoozbakht conjecture and would imply Cramer's conjecture. The very fact Cramer's conjecture pops up is enough to discourage any attempt of proof. Nonetheless, I shall raise this as an open question.

**Question**: More than the main conjecture, I am interested in the following special case question. Prove or disprove that

$$ \frac{\ln\Big(\frac{\sqrt[n]{p_n}}{\sqrt[n+1]{p_{n+1}}}-1\Big)}{\ln n} = -2 + \frac{\ln\ln n}{\ln n} + O\Big(+ \frac{\ln\ln n}{\ln^2 n}\Big). $$

Edit: Added the asymptotic form of the limit for primes.