Buchsbaumness conjecture for Stanley–Reisner rings with non-isolated singularities

Let $\Delta$ be a simplicial complex, let $\mathbbm{k}[\Delta]$ be its Stanley–Reisner ring, and let $\theta_1,\theta_2$ be the linear forms in the setting of Theorem annihilateThm. The singularities of $\Delta$ are homologically isolated when they satisfy the homological isolation property described in the surrounding discussion.

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

$$\mathbbm{k}[\Delta]/(\theta_1,\theta_2)\mathbbm{k}[\Delta]$$

is never Buchsbaum.

This conjecture proposes a converse to the known implication that homologically isolated singularities yield Buchsbaumness for the corresponding quotient in the relevant setting. The statement concerns examples where the quotient is quasi-Buchsbaum but not Buchsbaum, and the general assertion remains open.