Monotonicity of v_p(I^k) and v(I^k) across powers

Determine whether for any homogeneous ideal I in a standard graded polynomial ring S and for all p in the stable associated primes Ass∞(I), the inequalities v_p(I^k) < v_p(I^{k+1}) hold for all k ≥ 1, and whether v(I^k) < v(I^{k+1}) holds for all k ≥ 1.

Background

Although v_p(I^k) and v(I^k) are known to be eventually linear for the I-adic filtration, it is not known whether these sequences are strictly increasing from the outset. The authors present this as a concrete open question.

A positive answer would constrain the initial fluctuations of these sequences and complement the asymptotic linearity results.