Convergence of the ballistic-deposition growth speed constant

Prove the convergence of the rescaled ballistic-deposition growth speed $Kv_K$ as the strip size $K$ tends to infinity, where $v_K$ is the almost-sure limiting height-growth speed on a cyclic strip of $K$ sites.

Background

For ballistic deposition on a cyclic strip with $K$ sites, the paper proves that all sites share a common almost-sure growth speed $v_K$ and establishes the bounds $3/Kc v_Kc 3e/K$. Numerical evidence suggests that $Kv_K$ increases with $K$ toward a limit near $4.06$, while the earlier conjectural value $4/K$ is shown not to be exact. The equality of sitewise growth speeds is proved, but convergence of the rescaled constants $Kv_K$ remains unresolved.