Ahmed–Kullmann–Snevily's one-count bound for good partitions

For $t\geqslant 3$, let $S(t)$ be the set of good partitions considered in the paper, and for $B\in S(t)$ let $n_1(B)$ denote the number of $1$'s in $B$. One-count bound conjecture. For every $B\in S(t)$ and every $B'\in S(t-1)$,

$$n_1(B')\leqslant n_1(B)\leqslant w(2;3,t-1)+2t.$$

The claim is an observed property of good partitions in the computed range and is presented as a conjecture; no resolution is supplied in the paper excerpt.