Thomason's conjecture on balanced subdivisions

Given a graph $H$ and an integer $\ell$, a balanced $H$-subdivision is obtained by replacing every edge of $H$ with an internally vertex-disjoint path of length $\ell$, denoted $TH^{(\ell)}$. For a graph $G$, let $\delta(G)$ be its minimum degree. Thomason's conjecture. For every $k\ge 1$, there exists a function $f(k)$ such that if

$$\delta(G)\ge f(k),$$

then $G$ contains a balanced subdivision of $K_k$. The source states that Liu and Montgomery confirmed this conjecture. The result is part of the study of how minimum or average degree forces subdivisions with all replacement paths of equal length.