  We show the necessary part of the following theorem : a finitely generated,
residually finite group has property $PL^p$ (i.e. it admits a proper isometric
affine action on some $L^p$ space) if, and only if, one (or equivalently, all)
of its box spaces admits a fibred coarse embedding into some $L^p$ space. We
also prove that coarse embeddability of a box space of a group into a $L^p$
space implies property $PL^p$ for this group.
