Weak approximation by bounded Sobolev maps with values into complete manifolds

We have recently introduced the trimming property for a complete Riemannian manifold $N^n$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1,p}(B^m;N^n)$ when $p\in \{1,\dots ,m\}$. We prove in this note that, even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.