Does a positive sequence $\{a_n\}$ with $\limsup_{n} a_n^{1/n}=1$ and $\liminf_{n}a_n^{1/n} <1$ must have a subsequence $\{a_{n_i}\}$ satisfying $\lim_{i} a_{n_i}^{1/n_i}=1$ and $\lim_{i} |a_{n_i}^2-a_{n_i-1}a_{n_i+1}|^{1/n_i}=1$.
So Here are the hypothesis: \begin{equation} \limsup_{n} a_{n}^{1/n}=1 \tag1 \end{equation} and \begin{equation} \liminf_{n} a_{n}^{1/n}<1 \tag2 \end{equation} How to derive the conclusion: there is a subsequence $\{a_{n_i}\}$ fulfilling both of \begin{equation} \lim_{i} a_{n_i}^{1/n_i}=1 \tag3 \end{equation} and \begin{equation} \lim_{i} |a_{n_i}^2-a_{n_i-1}a_{n_i+1}|^{1/n_i}=1 \tag4 \end{equation}
My understanding to this problem
It is obvious that $(1)$ implies $(3)$, but how $(1)$ together with $(2)$ imply $(3)$ and $(4)$ is not so clear, although trivial examples such as $$1,\delta^2,1,\delta^4,\cdots,1,\delta^{2n},\cdots$$ (where $0<\delta<1$) strongly support this proposition.
Given $(3)$ holds, $(4)$ may be replaced with the equivalent \begin{equation}\lim_{i} \left|1-\frac{a_{n_i+1}}{a_{n_i}} \frac{a_{n_i-1}}{a_{n_i}}\right|^{1/n_i} =1 \end{equation}
So this suggests that $\dfrac{a_{n_i+1}}{a_{n_i}} \dfrac{a_{n_i-1}}{a_{n_i}}$ must be "small" enough. It seems somehow related to the ratio test vs the root test in convergence of series.
What I expect
A proof (or a counter-example) is of course appreciated, but I also want to know the origin of this problem (in what literature did it emerge). Discussions giving an insight to this problem is welcomed as well.