Liu–Montgomery's crux conjecture for clique subdivisions

Let $G$ be a graph and let $calpha=\tfrac{1}{100}$. For a subgraph $H\subset G$, write $d(H)$ for its average degree, and let $C_{\alpha}(G)$ be the minimum order of a subgraph $H\subset G$ with $d(H)\geq \alpha d(G)$. A subdivision of a clique is obtained by replacing each edge of the clique by internally vertex-disjoint paths.

Liu–Montgomery's crux conjecture. There exists some constant $c>0$ such that every graph $G$ contains a subdivision of a clique with at least

$$c\min \Big\{d(G), \sqrt{\tfrac{C_{\alpha}(G)}{\log C_{\alpha}(G)}}\Big\}$$

vertices.

The conjecture formalizes the idea that graphs without a small subgraph almost as dense as the whole graph should contain a large clique subdivision. The parameter $C_{\alpha}(G)$ is the crux, measuring the order of the smallest dense patch; the source states that Liu and Montgomery raised this conjecture with $\alpha=\tfrac{1}{100}$, but gives no resolution here.