Structure of minimal thick matroids

Characterize every minimal (t+1)-thick matroid as a tightening of the matroid described by the paper’s triangle construction.

Background

A matroid is defined to be (t+1)-thick when it is not the union of t hyperplanes. The paper discusses minimal thick matroids as extremal objects related to the covering problem and compares distinct extremal configurations that are nearly covered by t lines.

The conjecture proposes that all minimal (t+1)-thick matroids arise from the triangle-construction matroid by tightening, an operation described in the paper as adding a circuit on three points. The authors specifically identify the conjecture as an open structural problem even in the representable case.