The upper-bound conjecture for construction lines determined by registration marks

Let $k\geq 3$ registration marks be points in the plane, and consider the construction lines determined by the arrangement, subject to the two rules in Section 1. Upper-bound conjecture. At most $k+1$ construction lines are determined by the $k$ registration marks. The conjecture is motivated by the classification of near-pencils, augmented near-pencils, and railtracks, all of which determine at most $k+1$ construction lines; no arrangement with more than $k+1$ lines has been identified, but the claim remains open.