Half log-convexity and half log-concavity conjecture for ratios of Boros-Moll coefficients

Let $d_\ell(m)$ be the coefficient of $x^\ell$ in the Boros-Moll polynomial $P_m(x)$. For fixed $m$, consider the ratio sequence $d_\ell(m)/d_{\ell-1}(m)$. Ratio-shape conjecture. For $m\geq 3$, the sequence $\{d_\ell(m)/d_{\ell-1}(m)\}_{1\leq\ell\leq\lfloor m/2\rfloor+1}$ is log-convex, and the sequence $\{d_\ell(m)/d_{\ell-1}(m)\}_{\lfloor m/2\rfloor\leq\ell\leq m}$ is log-concave. Equivalently, the displayed strict inequality in the source holds for $2\leq\ell\leq\lfloor m/2\rfloor$, with the inequalities reversed for $\lfloor m/2\rfloor+1\leq\ell\leq m-1$. Numerical experiments in the source motivate this conjecture.