Maximal CM labelling conjecture for subdivided polygons

Let an $n$-gon have a subdivision with $k$ additional edges and no additional vertices. A CM monomial labelling is a monomial labelling whose cellular complex gives a Cohen–Macaulay cellular resolution, and it is maximal if it cannot be enlarged within the relevant class. Polygonal maximal-labelling conjecture. Any maximal CM monomial labelling of the subdivided polygon has $n+k$ variables. This predicts an exact variable count for maximal Cohen–Macaulay monomial labellings of polygonal subdivisions; no resolution or partial result is stated in the supplied text.