Asymptotic inequality between v-number and regularity for graded filtrations

Determine whether, for any Noetherian graded filtration I = {I[k]} of a finitely generated N-graded domain R, the v-number v(I[k]) is strictly less than the Castelnuovo–Mumford regularity reg(I[k]) for all sufficiently large integers k.

Background

The paper proves that for a Noetherian graded filtration I the v-function k ↦ v(I[k]) is eventually quasi-linear (Theorem 1.3). For many classes of filtrations, Castelnuovo–Mumford regularity is also known to be eventually linear/quasi-linear. The authors ask for a direct asymptotic comparison between these two invariants.

This question seeks to understand whether, in the asymptotic regime, the v-number is always dominated by the regularity for general Noetherian graded filtrations.