Random Groups at Density d<3/14 Act Non-trivially on a CAT(0) Cube Complex

For random groups in the Gromov density model at d<3/14, we construct walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities d<1/5 and d<5/24, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density d<1/4.