Boros and Moll's infinite log-concavity conjecture for Boros-Moll sequences

Let $d_\ell(m)$ denote the coefficient of $x^\ell$ in the Boros-Moll polynomial $P_m(x)$, for $0\leq \ell\leq m$. A sequence is $k$-log-concave when its first $k$ iterates under the operator $\mathcal{L}$ are nonnegative, where $\mathcal{L}(\{a_i\}_{i\geq 0})=\{a_i^2-a_{i-1}a_{i+1}\}_{i\geq 0}$ and $a_{-1}=0$; it is $\infty$-log-concave when this holds for every $k\geq 1$. Boros and Moll's conjecture. The Boros-Moll sequence $\{d_\ell(m)\}_{\ell=0}^m$ is $\infty$-log-concave. This conjecture remains open, despite the known log-concavity of these sequences.