On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called *log-concave* if $\mathcal{L}a_k\geq0$ for all $k\geq0$. 

One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called *doubly log-concave* if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:
>**QUESTION.** Suppose $(a_k)_k$ is doubly log-concave. Is the following true? If not, give a minimal added condition to secure it.
$$\frac{(\mathcal{L}a_k)^2}{a_k^2}
-\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\leq0, \qquad \text{for all $k\geq0$}.$$