Bj"orner–Swartz inequalities and M-sequence conjecture

Let $\Delta$ be a $(d-1)$-dimensional simplicial complex that is doubly Cohen–Macaulay over $k$. Write ${\mathfrak h}_i(\Delta)$ for its ${\mathfrak h}$-vector and ${\mathfrak g}(\Delta)$ for its ${\mathfrak g}$-vector. An $M$-sequence is a sequence satisfying the numerical conditions of a Hilbert function of a standard graded algebra. Bj"orner–Swartz conjecture. The following assertions hold:

$${\mathfrak h}_i(\Delta)\leq {\mathfrak h}_{d-i}(\Delta)$$

for $0\leq i\leq\lfloor d/2\rfloor$. 2.

$${\mathfrak g}(\Delta)$$

is an $M$-sequence.

The conjecture is proposed for doubly Cohen–Macaulay complexes as a numerical extension of the ${\mathfrak g}$-theorem. The source notes that the corresponding symmetry for general Buchsbaum* complexes fails, while these assertions remain open in the stated generality.