The concise degree-sequence graph-counting conjecture

Let $\mathbf d=(d_1,\ldots,d_n)\in\mathbb N^n$ satisfy $0\le d_i\le n-1$, and let $c(\mathbf d)$ be the number of simple graphs on vertices $v_1,\ldots,v_n$ with $\deg(v_i)=d_i$ for every $i$. Degree-sequence conciseness conjecture. For every $k>0$, there exists $\mathbf d$ such that $c(\mathbf d)=k$ and

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

for some fixed $C,c>0$. The decision problem for graphical degree sequences is polynomial-time decidable, while the conjectured concise value realization is open and can be viewed as a binary symmetric contingency-table problem.