Extended reverse ultra log-concavity conjecture for adjacent-ratio transposed Boros-Moll sequences

Let $d_\ell(m)$ be the coefficient of $x^\ell$ in the Boros-Moll polynomial $P_m(x)$. For each fixed $\ell\geq 0$, define the sequence $a_m=d_\ell(m-1)d_\ell(m+1)/d_\ell^2(m)$ for $m\geq\ell+1$. A sequence is extended reverse ultra log-concave according to the paper's definition of extended ultra log-concavity with all relevant inequalities reversed. Adjacent-ratio reverse ultra log-concavity conjecture. For each $\ell\geq 0$, the sequence $\{d_\ell(m-1)d_\ell(m+1)/d_\ell^2(m)\}_{m\geq\ell+1}$ is extended reverse ultra log-concave. This is proposed as a conjecture about the sharp asymptotic behavior of the transposed Boros-Moll sequences, and remains unresolved in the supplied text.