The weak Roberson conjecture states that any class 
of finite simple graphs which
excludes some finite simple graph and is closed under
graph minors and disjoint
union fails to distinguish some pair of nonisomorphic graphs
by graph homomorphism counts (Roberson 2026).
More precisely, there should exist nonisomorphic graphs 
for which
 |
where 
counts graph homomorphisms from 
to 
.
The conjecture remains open. The disproof of the strong Roberson conjecture does not refute this weaker statement (Kristjánsson 2026).
