We show that critical percolation on a product of two regular trees of degree ≥ 3 satisfies the triangle condition. The proof does not examine the degrees of vertices and is not “perturbative” in any sense. It relies on an unpublished lemma of Oded Schramm.
"Percolation on a product of two trees." Ann. Probab. 39 (5) 1864 - 1895, September 2011. https://doi.org/10.1214/10-AOP618