The asymptotic-direction conjecture for minimizing geodesics in Euclidean first-passage percolation

Let $B^{\mathrm{E}}_n=B^{\mathrm{E}}(0,n)$ be the Euclidean ball of radius $n$, let $\mathcal V_n\subseteq S^{d-1}$ be the set of initial velocities of minimizing unit-speed geodesics from the origin to the boundary sphere of $B^{\mathrm{E}}_n$, and let

$$\mathcal V=\bigcap_n\mathcal V_n.$$

For $v\in\mathcal V$, let $\gamma_v$ be the one-sided minimizing geodesic starting at the origin with initial velocity $v$. Asymptotic-direction conjecture. For every $v\in\mathcal V$, there exists $w\in S^{d-1}$ such that

$$\lim_{t\to\infty}\frac{\gamma_v(t)}{|\gamma_v(t)|}=w.$$

Howard and Newman proved the analogous assertion for a rotationally invariant Euclidean first-passage percolation model whose limiting shape is a Euclidean ball; the source conjectures it for the present model, where the corresponding result remains open.