Converse to the tropical-rank criterion for tropical bases

Let $L$ be an $(n-k)$-dimensional linear subspace of $K^n$ whose tropical Plücker coordinates are all finite, and let $M\in K^{m\times n}$, with $m\geq k$, have nonzero rows in the orthogonal complement of $L$. The rows of $M$ form a tropical basis for $L$ if and only if any $k$ columns of $\operatorname{deg}(M)$ have tropical rank $k$. Converse to the tropical-rank criterion. If any $k$ columns of $\operatorname{deg}(M)$ have tropical rank $k$, then the rows of $M$ form a tropical basis for $L$. The theorem preceding this conjecture proves the forward implication; the converse would give a criterion for a set of linear forms to form a tropical basis in the realizable case where all tropical Plücker coordinates are finite, generalizing the cited earlier result.