3 problems

The quadratic Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

Degree-two Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

Hirsch conjecture. If $\Delta$ is the boundary of a convex polytope, then $I_\Delta$ is Hirsch.