Ahmed–Kullmann–Snevily's double-zero conjecture 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 $n_{00}(B)$ denote the number of occurrences of $00$ in $B$. Double-zero conjecture. For every $B\in S(t)$,

$$n_{00}(B)<t.$$

The claim is based on the tabulated cases $3\leqslant t\leqslant 14$ and is presented as a conjecture; no resolution is supplied in the paper excerpt.