Boros–Moll infinite log-concavity conjecture for the Boros–Moll sequences

Let $P_n(x)=\sum_{i=0}^n d_i(n)x^i$ be the Boros–Moll polynomial, so that $\{d_i(n)\}_{i=0}^n$ is its coefficient sequence. A finite nonnegative sequence is infinitely log-concave if every iterate of the operator $\mathcal{L}(\{a_i\})=\{a_i^2-a_{i+1}a_{i-1}\}$, with $a_{-1}=a_{n+1}=0$, is nonnegative. Boros–Moll's conjecture. For every nonnegative integer $n$, the sequence $\{d_i(n)\}_{i=0}^n$ is infinitely log-concave. This conjecture was proved by Brändén; the ordinary log-concavity had earlier been proved by Kauers and Paule.