The structure of Grassmanian cluster categories

Grassmannian cluster categories were introduced in 2016 by Jensen--King-Su as categories of Cohen-Macaulay modules for certain algebras. Also in 2016, Baur-King-Marsh showed how these are related to dimer models on a disc. However, many questions remain about the structure and properties of these categories and how these are reflected in the dimer model. Amongst these are:

-Find a good combinatorial formula for the dimensions of Ext groups between modules, using the dimer model combinatorics.

-Find the relationship between a categorical formula for cluster characters introduced more recently by Jensen-King-Su and a combinatorial formula of Lam.

-Find a categorical interpretation of the quadratic form on the Grothendieck group of the category that seems (experimentally) to constrain the existence of indecomposable modules.