Non-triangular local resolution conjecture for repeated non-Desarguesian affine planes

Let $q$ be a prime power, and let a BIBD consist of $q$ copies of a non-Desarguesian affine plane of order $q$. A non-triangular local resolution system is a local resolution system that is not triangular. Non-triangular local resolution conjecture. Such a BIBD does not possess a non-triangular local resolution system. The claim is motivated by computational evidence for the first few values of $q$, where the corresponding examples arise from Desarguesian affine planes and no analogous system was found for non-Desarguesian planes; its general validity remains open.