Asymptotic equality of the v-function for maximal associated primes

Let $S=K[x_1,\dots,x_n]$ be a standard graded polynomial ring over an infinite field $K$, let $I\subset S$ be a homogeneous ideal, and let $\operatorname{Max}^\infty(I)$ denote the set of primes in the stable maximal-associated-prime set of the powers of $I$. For a prime $\mathfrak{p}$, let \v_{\mathfrak{p}}(I^k) be the corresponding local v-function, let $\alpha(I)$ be the initial degree of $I$, and let $\v(I^k)$ be the global v-function.

Asymptotic maximal-prime conjecture. For all $\mathfrak{p}\in\operatorname{Max}^\infty(I)$, one has

\lim_{k\rightarrow\infty}\frac{\v_{\mathfrak{p}}(I^k)}{k}=\alpha(I).

The paper motivates this claim using the eventual linearity of the global and local v-functions and the known equality $\lim_{k\rightarrow\infty}\v(I^k)/k=\alpha(I)$. It is presented as an expectation supported by experimental evidence; its resolution is not given in the supplied text.