Fibred coarse embeddability of box spaces and proper isometric affine actions on $L^p$ spaces

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.