High-dimensional critical percolation and critical Branching random walks are intersection equivalent

We prove that on Zᵈ, with $d>6$, and under the usual assumption that the two-point function and the Green's function are of the same order, the probabilities to hit any finite set for a critical Branching random walk starting from the origin and the critical percolation cluster of the origin are within a positive multiplicative constant one from each other, as long as the set is sufficiently far away from the origin comparatively to its diameter. The proof relies on the introduction of a truncated equilibrium measure in percolation and the derivation of an approximate last passage formula.