Admissible-subgraph density conjecture for partial subdivisions of complete graphs

Let $t\geq 3$, let $\mathcal{H}$ be the family of partial subdivisions of $K_t$, and let $(Q,\phi)$ be the weighted graph object used in the admissibility framework. An $H$-admissible subgraph is the admissible configuration defined in that framework for $H\in\mathcal{H}$, and $\phi(Q)$ denotes the corresponding total weight.

Admissible-subgraph density conjecture. If $(Q,\phi)$ has no $H$-admissible subgraph for any $H\in\mathcal{H}$, then

$$\phi(Q)\geq \frac{1}{t-1}.$$

The source says that proving this statement would suffice to prove the generalized weak-subdivision conjecture, because this is the only part of the preceding proof that does not apply to general families of graphs $\mathcal{H}$. The supplied context does not provide a resolution of this auxiliary conjecture.