Lovász's planar graph homomorphism isomorphism question

Let $H$ and $H'$ be two simple graphs, possibly with loops. For every planar multigraph $G$, possibly with multiple edges but no loops, consider the homomorphism counts $\hom(G,H)$ and $\hom(G,H')$.

Lovász's planar homomorphism question. If

$$\hom(G,H)=\hom(G,H')$$

for every such planar multigraph $G$, does it follow that $H$ and $H'$ are isomorphic?

The unrestricted analogue is a theorem of Lovász, but the corresponding statement when the input graphs are restricted to planar graphs was not known in the source. This question concerns whether planar homomorphism counts determine the target graph up to isomorphism.