Tropical-basis conjecture for minors of a generic matrix

Let $d$, $n$, and $r$ be positive integers, and consider a $d\times n$ matrix of variables. The $(r+1)\times(r+1)$ minors are polynomials in these variables, and a collection of polynomials is a tropical basis when its tropical hypersurfaces intersect in the tropical variety they define.

Tropical-basis conjecture. The $(r+1)\times(r+1)$ minors of a $d\times n$ matrix of variables are a tropical basis if and only if $r\leq 2$ or $r\geq \operatorname{min}(d,n)-2$.

This conjecture proposes a complete characterization of the sizes of minors that form tropical bases. The paper proves the case of the $4\times4$ minors of a $5\times n$ matrix for $n\geq4$, but the general assertion remains open.