Extremal Betti table generation conjecture for finite-length bigraded modules

Let $T$ be the bigraded polynomial ring under consideration, and consider the cone of Betti tables of bigraded $T$-modules with finite length. A module $M$ satisfies Claim~ when it has the property specified by that claim.

Extremal Betti table generation conjecture. All extremal Betti tables of the cone of bigraded $T$-modules with finite length are generated by Betti tables of modules $M$ that satisfy Claim~.

The conjecture proposes that the modules characterized by Claim~ account for every extremal ray of this cone. The preceding discussion gives a collection of extremal rays and notes that no condition is known for determining which $(1,1)$-valent connected graphs arise as matching graphs of modules.