Polynomial slowdown in an angle-dependent 2d branching Brownian motion

We consider a branching Brownian motion in ${\mathbb{R}}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(\theta )$ which depends only on the angle $\theta$ of the particle. We assume that $b$ is maximal when $\theta =0$, which is the preferred direction for breeding. Furthermore we assume that $b(\theta )=1-\beta \text{abs}{\theta }^{\alpha }+O({\theta }^2)$, as $\theta \to 0$, for $\alpha \in (2\mathrm{/}3,2)$ and $\beta >0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t){)}_{t\ge 1}$ is tight where $$m(t)=\sqrt{2}t-\frac{{\vartheta }_1}{\sqrt{2}}{t}^{(2-\alpha )\mathrm{/}(2+\alpha )}-(\frac{3}{2\sqrt{2}}-\frac{\alpha }{2\sqrt{2}(2+\alpha )})\log t\mathrm{.}$$ and ${\vartheta }_1$ is explicit in terms of the first eigenvalue of a certain operator.