The projective-dimension-two equality conjecture for Betti-diagram semigroups

Let $B_{\mathbb N}$ be the semigroup of virtual Betti diagrams and let $B_{\operatorname{mod}}$ be the semigroup of Betti diagrams of actual modules. A diagram has projective dimension $2$ when its associated module has projective dimension $2$.

Projective-dimension-two equality conjecture.

$$B_{\mathbb N}=B_{\operatorname{mod}}$$

for projective dimension $2$ diagrams.

This conjecture asserts that every virtual Betti diagram of projective dimension $2$ is realized by a module. The equality is known for projective dimension $1$ and for projective dimension $2$ level modules, but the general projective-dimension-two case remains open.