Quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster for $d\ge 1$

In this work we extend the quenched local limit theorem obtained by the authors in [BBDS23]. More precisely, we consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d\ge 1$ being the spatial dimension. In [BBDS23] an annealed local central limit theorem was proven for all $d\ge 1$ and a quenched local limit theorem under the assumption $d\ge 3$. Here we show that the latter result also holds for all $d\ge 1$.