Hilbert-series dimension bound for nine powers in six variables

Determine whether the lower bound d for the leading coefficient of the Hilbert series of R_{6,9,d} is sharp for every positive integer d.

Background

Proposition 6.9 proves that the Hilbert series of R_{6,9,d} has degree 3(d−1) and that its leading coefficient is at least d. The proof constructs d linearly independent elements in the relevant inverse system using two cubic forms and the structure of the CoxNagata ring. The authors report computational verification only through d=7 and leave the general sharpness of this bound unresolved.