Caterpillar symbic tree basis conjecture

Let $E$ be the ground set of matrix entries, and let $\mathcal{B}\subset E$ be a set of size $2n-1$. A symbic tree is a combinatorial object indexing a maximal cone in the tropical variety of rank-two symmetric matrices; a caterpillar symbic tree is a symbic tree whose branches are caterpillars, meaning that each branch contains only one cherry. Caterpillar symbic tree basis conjecture. The set $\mathcal{B}$ is a basis of the matroid of some symbic tree if and only if $\mathcal{B}$ is a basis of the matroid of some caterpillar symbic tree. This would reduce the determination of all bases in the relevant matroid to symbic trees with caterpillar branches; the preceding proposition establishes the analogous statement for the union of bases of the algebraic matroids, but the conjectured equivalence for each basis is not resolved here.