The toric g-vector conjecture for non-simplicial polytopes

A polytope has a toric $g$-vector, whose entries form a candidate for an $M$-sequence. An $M$-sequence is a sequence satisfying Macaulay inequalities, namely the inequalities referred to in the source.

Toric $g$-vector conjecture. The toric $g$-vector of a non-simplicial polytope is an $M$-sequence, i.e. it satisfies Macaulay inequalities.

This conjecture seeks an $f$-vector-type characterization for non-simplicial polytopes. The source motivates it by the desirability of an algebraic object and shifting operator for more general graded posets, but gives no resolution.