Bj\orner--Swartz conjecture on dual WLP for doubly Cohen--Macaulay complexes

Let

$$be a simplicial complex that is **doubly Cohen--Macaulay** over a field$$

, meaning that

$$is Cohen--Macaulay and$$

with any vertex removed remains Cohen--Macaulay of the same dimension. The dual WLP is the injectivity condition for the central multiplication map on the Artinian reduction of the Stanley--Reisner ring.

Bj\orner--Swartz conjecture. Every doubly Cohen--Macaulay simplicial complex has the dual WLP.

Doubly Cohen--Macaulayness is equivalent to being both Buchsbaum* and Cohen--Macaulay, and every proper link of a Buchsbaum* complex is doubly Cohen--Macaulay. The conjecture was cited as an open problem in the source.