Generalized conjecture on v-numbers of graded ideals with linear powers

Let $S$ be a standard graded polynomial ring and let $I\subset S$ be a graded ideal with linear powers, meaning that every power $I^k$ has a linear free resolution. Let $\alpha(I)$ be the initial degree of $I$, let $\operatorname{Ass}(I)$ be the set of associated primes of $I$, and for each $\mathfrak p\in\operatorname{Ass}(I)$ let $\alpha(\mathfrak p)$ denote the initial degree of $\mathfrak p$. Define

$$c(I):=\max\{\alpha(\mathfrak p)\mid \mathfrak p\in\operatorname{Ass}(I)\}.$$

Let $\mathrm{v}(I^k)$ denote the v-number of $I^k$. Generalized v-number conjecture. For all $k\geq 1$, one has

$$\mathrm{v}(I^k)=\alpha(I)k-c(I).$$

This statement generalizes Ficarra's monomial conjecture to arbitrary graded ideals with linear powers. The source explains that Ficarra's analogous assertion is false outside the monomial setting, motivating the correction by the term $c(I)$; the generalized statement remains conjectural here.