The 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. Large-clique conjecture. There are constants $c_1,c_2>0$ such that, for all integers $t\geq 3$, $k\geq c_1\log t$, and $n\geq t-1$,

$$\textup{cliques}(n,t,k)\leq (c_2t)^k n.$$

For fixed $k$, the quantity is known up to a constant factor of order $(t\sqrt{\log t})^{k-1}n$, while the conjectured bound is known for sufficiently large $k$, leaving the intermediate range open.