We consider integral functionals $F_\epsilon^{(j)}$, doubly indexed by $\epsilon$ > 0 and $j \in \mathbb{N} \cup \{\infty\}$, satisfying a standard linear growth condition. We investigate the question of $\Gamma$-closure, i.e., when the $\Gamma$-convergence of all families $\{F_\epsilon^{(j)}\}_ε$ with finite $j$ implies $\Gamma$-convergence of $\{F_\epsilon^{(\infty)}\}_ε$. This has already been explored for $p$-growth with $p$ > 1. We show by an explicit counterexample that due to the differences between the spaces $W^{1,1}$ and $W^{1,p}$ with $p$ > 1, an analog cannot hold. Moreover, we find a sufficient condition for a positive answer.