Realizing prescribed stability indices for depth, associated primes, regularity, and v-number

Determine whether there exists a graded ideal I in a polynomial ring S such that astab(I) = a, dstab(I) = d, rstab(I) = r, and vstab(I) = v for given positive integers a, d, r, v.

Background

The authors discuss several stability indices for graded ideals: astab(I) for stabilization of associated primes, dstab(I) for depth stabilization, rstab(I) for regularity stabilization, and vstab(I) for the v-number stabilization.

They show these indices can behave independently (e.g., vstab(I) can be smaller or larger than astab(I)), and ask whether arbitrary prescribed values can be simultaneously realized.