A question about a blue fan and a red fan and their common refinement
Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $M$, a blue vertex $v$ of $M_1$, a red vertex $u$ of $M_2$ and two faces $F,G$ of $M$ such that 1) $v,w \in F$ , 2) $u,w \in G$, and 3) $\dim F +\dim G =n$.
A simple (but perhaps not the most general) setting in which to ask this question is with regard to the red and blue maps coming from red and blue polyhedral fans associated to red and blue convex polytopes. The common refinement will be the fan obtained by taking all intersections of cones, one from the first fan and one from the second.
For $n=2$ was proved by Paco Santos,Tamon Stephen and Hugh Thomas. They gave two proofs. One is based on an Euler characteristic argument, and the other applies a connectivity argument. Here is a link to the paper.
A positive answer to the conjecture in all dimensions will imply an upper bound of the form $nd$ to diameters of graphs of simple $d$-dimensional polytopes with $n$ facets. (Proving the conjecture for the simple case mentioned above would suffice.) This will be great as no polynomial upper bound is known. For more information see this post.