4 problems

Existence conjecture. The following conditions are equivalent:

Buchsbaumness conjecture. If the singularities of $\Delta$ are not homologically isolated, then

The authors' conjecture. If $e_1(Q)$ is independent of the choice of parameter ideal $Q$ of $A$, then $A$ is furthermore a Buchsbaum ring.

Sapko's annihilator conjecture. If $\operatorname{gr}_{\mathfrak m}(R)$ is Buchsbaum, then for some $k\geq 1$,