Infinite log-concavity conjecture for the sparse-polynomial sequence

Let $f_n(z)=f_{2,n}(z)$, and for a sequence $(a_n)$ define the operator

$$\mathcal L(a_n)=a_n^2-a_{n-1}a_{n+1}.$$

A sequence is called $k$-log-concave when $\mathcal L^j(a_n)\geq 0$ for $j=0,1,\ldots,k$, and infinite log-concavity conjecture. The sequence $(f_n(z))$ is $\infty$-log-concave for all $0\leq z\leq 1$. This is motivated by computations showing that repeated applications of $\mathcal L$ preserve nonnegative coefficients; the conjecture concerns all iterations and remains unresolved in the supplied source.