This asymptotic condition is essentially the weakest that can prevent the almost calibrated Lagrangian in a given homology class from escaping to infinity. For instance, if $L$ is a special Lagrangian in $X$, then $L\times S^{1}$ is a special Lagrangian in the product $X\times T^{*}S^{1}$, which can obviously be translated in the $\mathbb{R}$ direction to escape to infinity, albeit not preserving the $D^{b}Fuk(X)$ class. One can still hope to obstruct escaping to infinity using Floer theory, but then incorporating singular Lagrangians requires more foundational work.