Huang–Sudakov's biclique partition–chromatic number conjecture

Let $G$ be a graph. Its biclique partition number is the minimum number of bicliques, that is, complete bipartite graphs, needed to partition the edge set of $G$. Its chromatic number $\chi(G)$ is the minimum number of stable sets in a partition of the vertex set of $G$.

Huang–Sudakov's conjecture. For each integer $k>0$, there exists a graph $G$ with biclique partition number $k$ and chromatic number at least

$$2^{c \log^2 k},$$

for some constant $c>0$.

If true, this graph-theoretical statement would improve the known lower bound on the nondeterministic communication complexity of the clique-versus-stable-set problem from $\frac{6}{5}\log n-O(1)$ to $\Omega(\log^2 n)$. The exact deterministic and nondeterministic communication complexities remain unknown.