On the notion of positive rank
This is not an answer, but another version, NOT equivalent to the original of this excellent, but perhaps hard to grasp question.
Define the bipartite clique partition number of a graph as the least number of its complete bipartite subgraphs that cover each edge exactly once.
Similarly define the support of the fractional bipartite clique partition number of a graph as the least number of its complete bipartite subgraphs that have a positive weighing such that for each edge the sum of the weights of the subgraphs containing it is exactly one.
The question is whether these two parameters are always equal or not.

I find it quite surprising that while fractional version are studied a lot, I've never heard of studying the size of the support of the fractional version.
I also wonder whether this latter parameter has any applications.
Also, the question could be asked about several other parameters.