Slope of v_p(I^k) equals α(I) for maximal stable associated primes

Prove that for any homogeneous ideal I ⊂ S and for every p maximal in the stable set of associated primes Ass∞(I), the limit lim_{k→∞} v_p(I^k)/k equals α(I), the initial degree of I.

Background

It is known that lim_{k→∞} v(I^k)/k = α(I). Since v(I^k) = max_{p∈Ass∞(I)} v_p(I^k) for large k and each v_p(I^k) is eventually linear, at least one stable associated prime achieves slope α(I).

The conjecture strengthens this by asserting that every maximal element of Ass∞(I) has asymptotic slope equal to α(I).