Roberson's weak conjecture for bounded Hadwiger number

For a graph $G$, its Hadwiger number is the largest integer $t$ such that $G$ contains the complete graph $K_t$ as a minor. Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in that class. Roberson's weak conjecture. For every $k \geq 0$, there exist non-isomorphic graphs $G$ and $H$ that are homomorphism indistinguishable over all graphs of Hadwiger number $\leq k$. This is a concrete form of the question whether sparse graph classes can distinguish all non-isomorphic graphs. The paper proves the conjecture for vortex-free graph classes, while the general minor-excluded case remains unresolved.