Small $K_5$-models at average degree greater than six

Let $G$ be a graph, let $|G|$ denote its number of vertices, and let a $K_5$-model be a model of the complete graph $K_5$ in $G$. For a real number $\varepsilon>0$, suppose that the average degree of $G$ is at least $6+\varepsilon$. Small $K_5$-model conjecture. There is a function $h$ such that every such graph $G$ contains a $K_5$-model with at most

$$h(\varepsilon)\cdot\log|G|$$

vertices. The conjecture would improve the paper's bound from average degree at least $16+\varepsilon$ to the threshold $6+\varepsilon$; the case $t=5$ is identified as open, while the threshold $6$ for forcing a $K_5$-minor follows from Wagner's characterization of graphs with no $K_5$-minor.