Refined list Hadwiger's conjecture

Let $\lambda$ be a multiset of positive integers, let $k_\lambda$ be the sum of its elements, and let $|\lambda|$ be its number of elements. For a graph $G$, define $h(\lambda)$ to be the maximum integer $t$ such that every $K_t$-minor-free graph is $\lambda$-choosable. Refined list Hadwiger's conjecture. There are functions $\phi,\psi:\mathbb{N}\to\mathbb{N}$ such that

$$\lim_{n\to\infty}\psi(n)=\infty$$

and, for any multiset $\lambda$ of positive integers, if $k_\lambda\geqslant\phi(k_\lambda-|\lambda|)$, then

$$k_\lambda-h(\lambda)\geqslant\psi(k_\lambda-|\lambda|).$$

The quantity $k_\lambda-|\lambda|$ measures the distance between $\lambda$-choosability and ordinary $k_\lambda$-colourability. The conjecture asserts that, once $k_\lambda$ is sufficiently large relative to this distance, the gap between $k_\lambda$ and $h(\lambda)$ must grow without bound. The paper proves this for several families of multisets, but whether it holds for all $\lambda$ remains open.