The finiteness conjecture for $k$-variable TN obstructions

A TN obstruction is a pattern that is not TN-completable, where TN-completable means that every partial totally nonnegative matrix with the pattern admits a totally nonnegative completion. A pattern is $k$-variable if it has variability $k$.

TN-obstruction finiteness conjecture. For every $k$, there are finitely many $k$-variable TN obstructions.

The paper classifies all one-variable TN obstructions and proves finiteness in that case. Finiteness for arbitrary $k$ remains open.