Infinite strict log-concavity conjecture for transposed Boros-Moll sequences

Let $d_\ell(m)$ be the coefficient of $x^\ell$ in the Boros-Moll polynomial $P_m(x)$, and consider, for fixed $\ell$, the transposed sequence $\{d_\ell(m)\}_{m\geq\ell}$. Let $\mathcal{L}$ be the operator on sequences defined by $\mathcal{L}(\{a_i\}_{i\geq 0})=\{a_i^2-a_{i-1}a_{i+1}\}_{i\geq 0}$, with $a_{-1}=0$. A sequence is $\infty$-strictly-log-concave if every iterate $\mathcal{L}^j$ is strictly positive. Transposed Boros-Moll conjecture. The transposed Boros-Moll sequences $\{d_\ell(m)\}_{m\geq\ell}$ are $\infty$-strictly-log-concave for any $\ell\geq 3$. Numerical experiments support the conjecture, while the cases $\ell=0,1,2$ exhibit obstructions described in the source.