Edge-disjoint caterpillar realization conjecture for tree degree matrices

A degree matrix $D$ consists of rows that are degree sequences on the same set of $n$ labeled vertices; it is a tree degree matrix when every row is a tree degree sequence, and it is without common leaves when each vertex is a leaf in at most one row. A caterpillar realization is an edge-disjoint realization in which every graph realizing a row is a caterpillar.

Caterpillar realization conjecture. Every tree degree matrix without common leaves has a caterpillar realization.

This conjecture strengthens the paper's equivalent formulation that an arbitrary number of tree degree sequences have edge-disjoint caterpillar realizations whenever every vertex is a leaf in at most one tree. The supplied text gives no resolution, so its status remains open.