Solid $k$-minor essentiality conjecture

Let a solid $k$-minor be a $k\times k$ minor using consecutive rows and consecutive columns, and call a minor $k$-essential if it is necessary in a $k$-positivity test in the sense developed in the paper. Solid-minor essentiality conjecture. Solid $k$-minors are $k$-essential. The conjecture is motivated by the preceding construction of $k$-positivity tests and the observed prevalence of solid $k$-minors; proving it would identify a distinguished class of minors that every such test must include.