Maximum segmentation conjecture for finite vector subtraction

Let $k\in\mathbb N$, and let a ruleset have size $k$ when it contains $k$ moves. An $r$-segmentation is an outcome segmentation consisting of $r$ outcome segments.

Maximum segmentation conjecture. For each $k\in\mathbb N$, $k\ne 2$, there exists a ruleset of size $k$ that has a $(k+1)$-segmentation. For each $k\in\mathbb N$, there is no ruleset of size $k$ that has a $(k+2)$-segmentation.

This conjecture proposes both attainability of the maximum observed number of segments and an upper bound in terms of the ruleset size. It is stated in the paper’s open-problems section and no resolution is supplied.