Rigid balanced point-line configurations

Construct families of rigid balanced $v_k$-configurations of points and lines for every relevant $k>3$.

Background

A balanced $v_k$-configuration has $v$ points, $v$ lines, and $k$ incidences at every point and line. The paper observes that the count forces flexibility for $k=3$ when $v>8$, and asks whether rigid families exist for larger valencies.