The finer first-passage-time asymptotic conjecture for branching random walks

Let $B_x$ be the shifted unit ball at Euclidean distance $x$ from the origin, let $\tau_x$ denote the first passage time of the branching random walk to $B_x$, and let $\widehat{c}_1$ and $I$ be the constants and rate function defined under assumptions (A1)--(A4), or under (A1), (A2), (A5), and (A6). For $d\geqslant 2$, conditioned upon survival, the finer first-passage-time asymptotic conjecture.

$$\tau_x=A(x)+O_\mathbb{P}(1)=\frac{x}{\widehat{c}_1}+\frac{d+2}{2\,\widehat{c}_1\partial_{x_1}I(\widehat{c}_1,\mathbf{0})}\,\log x+O_\mathbb{P}(1),$$

where the $O_\mathbb{P}(1)$ term is tight. This conjecture refines the established linear-plus-logarithmic first-passage-time asymptotic for branching random walks, analogously to the finer asymptotic known for branching Brownian motion; the claimed tightness of the remainder remains open.