The growth conjecture for TP obstructions

A TP obstruction is a pattern that is not TP-completable, where TP-completability means that every partial totally positive matrix with the pattern admits a totally positive completion. The number of rows of a pattern is its height.

TP-obstruction growth conjecture. There are infinitely many TP obstructions, and the number of TP obstructions with a given number of rows grows quickly, perhaps exponentially, with the number of rows.

The conjecture is motivated by the discovery of 78 new obstructions among the $4$-by-$4$ patterns, a number six times the total number of obstructions for all $3$-by-$n$ patterns. The paper provides no resolution of the asserted infinitude or growth rate.