The minimal morphism conjecture for regular pairs of weighted simplicial complexes

Let $(\overline{d},\overline{a}) \in \mathcal{CS}$ be a regular pair, and let $(\mathcal{D},\mathcal{A})$ be the associated pair of weighted simplicial complexes. Write $\operatorname{Vert}(\mathcal{A})$ and $\operatorname{Vert}(\mathcal{D})$ for their vertex sets. Minimal morphism conjecture. There exists a minimal morphism

$$\varphi: \operatorname{Vert}(\mathcal{A}) \rightarrow \operatorname{Vert}(\mathcal{D}).$$

This is proposed as a combinatorial approach to the existence of nice nef partitions and to the paper's main conjecture. The supplied text does not state whether the assertion has been resolved.