Summary: 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 (sufficiency is due to [X. Chen et al., Bull. Lond. Math. Soc. 45, No. 5, 1091–1099 (2013; Zbl 1347.20043)]). 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.