Closure result for \( \Gamma \)-limits of functionals with linear growth

We consider integral functionals \( \mathcal {F}^{(j)}_{\varepsilon } \), doubly indexed by \( \varepsilon > 0 \) 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 \( \{ \mathcal {F}^{(j)}_{\varepsilon } \}_{\varepsilon }\) with finite j implies \( \Gamma \)-convergence of \(\{ \mathcal {F}^{(\infty )}_{\varepsilon } \}_{\varepsilon }\). 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.