The reduction-number conjecture for asymptotic fullness

Let $(R, \mathfrak{m})$ be a Cohen–Macaulay local ring with infinite residue field and

$$\operatorname{dim} R\geq 2.$$

For a minimal reduction $I$ of $\mathfrak{m}$, let $n_3(I)$ denote the asymptotic fullness invariant defined in the paper, and let $\operatorname{r}_I(\mathfrak{m})$ be the reduction number of $\mathfrak{m}$ with respect to $I$. The reduction-number conjecture. For every minimal reduction $I$ of $\mathfrak{m}$,

$$n_3(I)=\operatorname{r}_I(\mathfrak{m}).$$

The conjecture is motivated by the authors' computations and by the dimension-one case, where the corresponding formula is independent of the chosen minimal reduction. It is proposed because no example was known violating the related inequality $s(\mathfrak{m})\leq \operatorname{r}_I(\mathfrak{m})+1$; the source provides no resolution in dimensions at least two.