Algebraic characterizations corresponding to constructibility and shellability of the lcm-lattice

Determine the algebraic property (or properties) of a monomial ideal I in a polynomial ring k[x1, ..., xn] that are equivalent to the constructibility of its lcm-lattice L(I), and determine the algebraic property (or properties) of I that are equivalent to the shellability of L(I).

Background

The paper establishes combinatorial-topological characterizations of two important algebraic properties of monomial ideals via the lcm-lattice: a monomial ideal has a d-linear resolution if and only if L(I) is d-degree graded and Cohen–Macaulay; and a monomial ideal generated in degree d has linear quotients if and only if L(I) is d-degree graded and CL-shellable.

Constructibility and shellability are topological/poset-theoretic properties that lie between Cohen–Macaulayness and CL-shellability in strength. The authors ask for corresponding algebraic characterizations on the ideal side that mirror their established equivalences for linear resolutions and linear quotients, seeking precise algebraic conditions on I that are equivalent to L(I) being constructible and to L(I) being shellable.