Triangle-free Rank--Ramsey graphs from ordered incidence lifts

For a graph $G$ with adjacency matrix $A_G$ and a coefficient field $\mathbb K$, put $ρ_{\mathbb K}(G)=\operatorname{rank}_{\mathbb K}(A_G+I)$. We answer the triangle-free Rank--Ramsey existence question of Beniamini, Linial and Shraibman. For every order $n$, we construct a triangle-free graph $H_n