Dimers from Postnikov diagrams

Question

I am reading several papers on Postnikov diagrams, dimer models, and quivers. From the Postnikov diagram, we can draw a dimer model and an ice quiver. My question is as follows.
Does every dimer model(or ice quiver) embedded in the disk have a corresponding Postnikov diagram?
I guess this is false but I cannot find a counterexample. Can anyone help?

Answer 1

Dimer models and plabic graphs both give you every representable matroids (so all positroids). So you can always (up to equivalence) find a reduced plabic graph that gives the same positroid.

But this result is way too overkill, if you check thoroughly the two definitions, you can see that they define the same objects.

This paper by Çanakçi, King, and Pressland can give you further information:
https://arxiv.org/pdf/2106.15924.pdf
The introduction also answers your question.