Worst-case $T^{1/3}$ correction conjecture for time-inhomogeneous branching Brownian motion

For one-dimensional branching Brownian motion with a smooth diffusivity profile $\sigma:[0,1]\to(0,\infty)$ whose image is contained in a compact subset of $(0,\infty)$, let $v_\sigma$ be the velocity determined by the paper's equation (eq-vsn13a), and let $g_\sigma(T)$ denote the correction term in the displacement asymptotic (eq-dispn13). Worst-case correction conjecture. The displacement asymptotic holds with this $v_\sigma$ and

$$|g_\sigma(T)|=O(T^{1/3}).$$

The authors motivate the conjecture by observing that strictly decreasing diffusivity appears to produce a correction of order $T^{1/3}$, rather than the logarithmic correction of homogeneous branching Brownian motion; they propose that this is the worst possible correction order. The conjecture is not resolved in the supplied text.