Realizing prescribed asymptotics for v-number and regularity of monomial ideals

Determine whether, given integers a ≥ 1, b ≥ −1, c ≥ 1, d ≥ 0 with ak + b ≤ ck + d for all sufficiently large k, there exists a monomial ideal I in a polynomial ring S such that v(I^k) = ak + b and reg(I^k) = ck + d for all sufficiently large k.

Background

For monomial ideals I, the v-number satisfies v(I^k) = a k + b for large k with a = α(I) ≥ 1 and b ≥ −1, and the regularity satisfies reg(I^k) = c k + d for large k with c ≥ 1 and d ≥ 0.

The question asks whether one can simultaneously prescribe these two eventual linear behaviors (subject to ak + b ≤ ck + d asymptotically) by choosing an appropriate monomial ideal.