The very-large-clique conjecture for graphs excluding a complete minor

Let $\textup{cliques}(n,t,k)$ denote the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor. Very-large-clique conjecture. There is some $\lambda\in[\tfrac13,1)$ such that, for all integers $t\geq 3$, $k\geq\lambda t$, and $n\geq t-1$,

$$\textup{cliques}(n,t,k)\leq \binom{t-2}{k-1}\left(n-\frac{(k-1)(t-1)}{k}\right).$$

This conjecture asserts that the lower bound established earlier is tight for cliques whose size is a fixed positive fraction of $t$. The source presents it as open; it concerns the extremal regime of very large cliques in $K_t$-minor-free graphs.