Bounded front-width conjecture for branching Brownian motion with mass decay

Let $d(t,\b7)$ and $D(t,\b7)$ denote the lower and upper front locations for branching Brownian motion with mass decay, respectively. For $\b7<1$, almost surely,

$$\limsup_{t\to\infty}\left|\inf_{s\geq 0}d(t+s,\b7)-D(t,b7)\right|<\infty.$$

Bounded front-width conjecture. For $b7<1$, almost surely,

$$\limsup_{t\to\infty}\left|\inf_{s\geq 0}d(t+s,b7)-D(t,b7)\right|<\infty.$$

The preceding theorem proves a weaker logarithmic bound, showing that the lower front catches up with the upper front after a delay of order $b7\log t$. The conjecture asserts that this delay remains uniformly bounded in the relevant asymptotic sense.