The concise contingency-table realization conjecture

Let $\mathbf a\in\mathbb N^r$ and $\mathbf b\in\mathbb N^s$ have common size $n=|\mathbf a|=|\mathbf b|$, and let $\operatorname{CT}(\mathbf a,\mathbf b)$ be the number of contingency tables with these margins. Contingency-table conciseness conjecture. For every $k>0$, there exist vectors $\mathbf a,\mathbf b$ of size $n$ such that

$$\operatorname{CT}(\mathbf a,\mathbf b)=k$$

and

$$n\le C(\log k)^c$$

for some fixed $C,c>0$. This asks for short unary-margin representations of every positive counting value and is open; the paper notes consequences for related counting functions.