Rigidity conjecture for Hirzebruch–Kummer coverings of rigid line configurations

Let ${\mathcal L}$ be a rigid line configuration, and let $HK(n, {\mathcal L})$ be the associated Hirzebruch–Kummer covering of exponent $n$. Rigidity conjecture. If ${\mathcal L}$ is rigid, then the surface $HK(n, {\mathcal L})$ is rigid for $n$ sufficiently large. This conjecture asks whether rigidity of the line configuration eventually implies rigidity of its Hirzebruch–Kummer coverings; it is known for the complete quadrangle, where $HK(n, {\mathcal C}{\mathcal Q})$ is rigid if and only if $n\geq 4$, but the general case remains open.