The uniform Erdős–Pósa conjecture for wheel models

Let $W_t$ denote the wheel on $t+1$ vertices, and let $t\geq 3$. For a graph $G$, a wheel model is a model of $W_t$ in $G$.

Uniform wheel-model Erdős–Pósa conjecture. There are a constant $c>0$ and a function $g:\mathbb{N}\to\mathbb{N}$ such that, for every integer $t\geq 3$, wheel models in $G$ have the Erdős–Pósa property with bounding function

$$c k\log k+g(t)k.$$

The paper proves an $O(k\log k)$ bound for each fixed wheel, while this conjecture asks that the coefficient of the logarithmic term be independent of the wheel size $t$. It is motivated by the analogous result for cycle models and remains open in the source.