Infinite log-monotonicity conjecture for transposed Boros-Moll sequences

Let $d_\ell(m)$ be the coefficient of $x^\ell$ in the Boros-Moll polynomial $P_m(x)$. For a sequence $\{a_i\}_{i\geq 0}$, let $\mathcal{R}\{a_i\}_{i\geq 0}=\{a_{i+1}/a_i\}_{i\geq 0}$. A sequence is infinitely log-monotonic when every iterate has alternating log-concavity and log-convexity as specified by the paper's definition. Infinite log-monotonicity conjecture. The transposed Boros-Moll sequence $\{d_\ell(m)\}_{m\geq\ell}$ is infinitely log-monotonic for $\ell=0$. The sequence $\{d_\ell(m+1)/d_\ell(m)\}_{m\geq\ell}$ is infinite log-monotonic for each $\ell\geq 1$. The $ll=0$ sequence is known to be log-convex and ratio log-concave, but the asserted infinite behavior remains open.