Star-system conjecture for 0-1 systems of parameters

Prove that whenever the edge-ring quotient S/I(G) of a graph G has a 0-1 system of parameters, it also has a star-system of parameters associated with some maximal independent set of G.

Background

The paper studies linear systems of parameters for quotients of monomial ideals, with particular emphasis on systems whose coefficient matrices have entries in {0,1}. For graph edge ideals, the authors define a star-system of parameters as a 0-1 system whose forms are supported on stars centered at the vertices of a specified maximal independent set.

The paper establishes the existence of star-systems for several classes of graphs, including König graphs, odd cycles, complements of cycles, and certain alpha-zero-reducible graphs. However, the authors also give an example showing that a 0-1 system need not be a star-system associated with every prescribed maximal independent set. They leave unresolved whether some maximal independent set always supports such a star-system whenever any 0-1 system exists.