Multi-separation star refinement conjecture

Let $(\vec{U},\leq,*,\vee,\wedge)$ be a distributive universe of separations with an order function, and let $\{\overleftarrow{x}_i:i\in[n]\}$ be a set of separations in $\vec{U}$. A set of separations is a star if its oriented separations point pairwise towards one another in the standard sense, and $\vec{u}_i$ is linked to $\vec{x}_i$ when the two separations satisfy the linking relation used in the preceding lemma.

Multi-separation star refinement conjecture. There exists a set of separations $\{\overleftarrow{u}_i:i\in[n]\}$ such that:

• $\{\overleftarrow{u}_i:i\in[n]\}$ is a star;
• $\vec{u}_i$ is linked to $\vec{x}_i$ for all $i\in[n]$;
• $|\overleftarrow{u}_i|\leq|\overleftarrow{x}_i|$ for all $i\in[n]$;
• $\overleftarrow{u}_i=\overleftarrow{x}_i\bigwedge_{j\neq i}\vec{u}_j$ for all $i\in[n]$.

This would extend the preceding two-separation lemma to an arbitrary finite set of separations. The text calls it tempting rather than established, and no proof or resolution is supplied here.